On the Classification of Irreducible Representations of Real Algebraic Groups*

R. P. Langlands

Introduction. Suppose G is a connected reductive group over a global field F. Many of the problems of the theory of automorphic forms involve some aspect of study of the representation ρ of G(A(F)) on the space of slowly increasing functions on the homogeneous space G(F)\G(A(F)). It is of particular interest to study the irreducible constituents of ρ. In a lecture [9], published some time ago, but unfortunately rendered difficult to read by a number of small errors and a general inprecision, reflections in part of a hastiness for which my excitement at the time may be to blame, I formulated some questions about these constituents which seemed to me then, as they do today, of some fascination. The questions have analogues when F is a local field; these concern the irreducible admissible representations of G(F).

As I remarked in the lecture, there are cases in which the answers to the questions are implicit in existing theories. If G is abelian they are consequences of class field theory, especially of the Tate-Nakayama duality. This is verified in [10]. If F is the real or complex field, they are consequences of the results obtained by Harish-Chandra for representations of real reductive groups. This may not be obvious; my ostensible purpose in this note is to make it so. An incidental, but not unimportant, profit to be gained from this exercise is a better insight into the correct formulation of the questions.

Suppose the F is the real or complex field. Let Π(G) be the set of infinitesimal equivalence classes of irreducible quasi-simple Banach space representations of G(F) [16]. In the second section we shall recall the definition of the Weil group W_{F} of F as well as that of the associated or dual group G^{∧} of G and then introduce a collection Φ(G) of classes of homomorphisms of the Weil group of F into G^{∧}. After reviewing in the same section some simple properties of the associate group we shall, in the third section, associate to each φ ∈ Φ (G) a nonempty finite set Π_{φ} in Π(G). The remainder of the paper will be devoted to showing that these sets are disjoint and that they exhaust Π(G). For reasons stemming from the study of L-functions associated to automorphic forms we say that two classes in the same Π_{φ} are L-indistinguishable.

Thus if Π˜(G) is the set of classes of L-indistinguishable representations of G(F), then by definition the elements of Π˜(G) are parametrized by Φ(G). It will be seen that if G is quasi-split and G_{1} is obtained from it by the inner twisting ψ then ψ defines an injection Φ(G_{1}) ↪ Φ(G) and hence an injection Π˜(G_{1}) ↪ Π˜(G).

* Preprint, Institute for Advanced Study, 1973. Appeared in Math. Surveys and Monographs, No. 31, AMS (1988)

It will also be seen that for G quasi-split the set Π˜(G) is, in a sense to be made precise later, a covariant function of G^{∧}. These properties of Π˜(G) provide answers to the questions of [9].

The classification of L-indistinguishable representation throws up more questions than it resolves, since we say nothing about the structure of the sets Π_{φ} themselves and hence do not really classify infinitesimal equivalence classes. Nonetheless we do reduce the general problem to that of classifying the tempered representations. This is a considerable simplification. For example, Wallach [15] has proved that the unitary principal series are irreducible for complex groups. From this it follows that each Π_{φ} consists of a single class; so the classification is complete in this case. Since Φ(G) may, when F is complex, be easily identified with the orbits of the Weyl group in the set of quasi-characters of a Cartan subgroup G(C), it likely that the classification provided by this paper coincides with that of Zhelobenko. The set Π(G) has been described by Hirai [7, 8] for G = SO(n, 1) or SU(n, 1). It is a simple and worthwhile exercise to translate his classification into ours. In fact, the definitions of this paper were suggested by the study of his results. It would be interesting to know if each Π_{φ} consists of a single class when G is GL(n) and F is R.

Important though these problems are, we do not try to decide which elements of which Π_{φ} are unitary or how the classes in a Π_{φ} are unitary or how the classes in a Π_{φ} decompose upon restriction to a maximal compact subgroup of G(R).

The three main lemmas of this paper are Lemmas 3.13, 3.14, and 4.2. The first associates to each triplet consisting of a parabolic subgroup P over R, a tempered representation of a Levi factor of P (R), and a positive quasi-character of P (R) whose parameter lies in the interior of a certain chamber defined by P, an irreducible quasi-simple representation of G(R). The second lemma shows that these representations are not infinitesimally equivalent. The third shows that they exhaust the classes of irreducible quasi-simple representations.

As we observed above, the proofs are not very difficult. Unfortunately, they rely to some extent on unpublished results of Harish-Chandra. To prove that the sets Π_{φ} are disjoint we use results from [6], which includes no proofs. Moreover, and this is more serious, for the proof of Lemma 4.2 we use results from [5], which has only been partly reproduced in Appendix 3 of [16]. It contains theorems on differential equations which are used to study the asymptotic behavior of spherical functions not only in the interior of a positive Weyl chamber, as in [16], but also on the walls.

2. The associate group. We begin by recalling some of the constructions of [9]. If F is C the Weil group W_{F} is C^{×}. If F is R the Weil group W_{F} consists of pairs (z, τ), z ∈ C^{×}, τ ∈ 𝔤(C/R) = {1, σ} with multiplication defined by

(z_{1}, τ_{1})(z_{2}, τ_{2}) = (z_{1}τ_{1}(z_{2})a_{τ_{1},τ_{2}}, τ_{1}τ_{2}).

Here a_{τ_{1}τ_{2}} = 1 if τ_{1} = 1 or τ_{2} = 1 and a_{τ_{1}τ_{2}} = −1 if τ_{1} = τ_{2} = σ. For both fields we have an exact sequence

1 → \overline{F}^{×} → W_{F} → 𝔤(\overline{F} /F) → 1.

Suppose Gô is a connected reductive complex algebraic group, Bô a Borel subgroup of Gô, and T ô a Cartan subgroup of Gô in Bô. For each root α^{∧} of T ô simple with respect to Bô let X_{α^{∧}} ≠ 0 in the Lie algebra 𝔤^{∧} of Gô be such that

Adt(X_{α^{∧}}) = α(t)X_{α^{∧}}, t ∈ T ô.

Let

A(Gô, Bô, T ô, {X_{α^{∧}}})

be the group of complex analytic automorphisms ω of Gô leaving Bô and T ô invariant and sending X_{α^{∧}} to X_{ωα^{∧}}, where ωα^{∧} is defined by

ωα^{∧}(ωt) = α^{∧}(t).

If instead of Bô, T ô, \{X_{α^{∧}}\} we choose \overline{B}ô, \overline{T} ô, \{\overline{X}_{\overline{α}}^{∧}\} with the same properties there is a unique inner automorphism ψ such that

\overline{B}ô = ψ(Bô); \overline{T} ô = ψ(T ô), \overline{X}_{ψα^{∧}} = ψ(X_{α^{∧}}).

Then

A(Gô, \overline{B}ô, \overline{T} ô, \{\overline{X}\overline{α}^{∧}\}) = \{ψωψ^{−1}|ω ∈ A(Gô, Bô, T ô, \{X_{α}^{∧}\})\}.

Suppose we have an extension

1 → Gô → G^{∧} → W_{F} → 1

of topological groups. A splitting is a continuous homomorphism from W_{F} to G^{∧} for which the composition

W_{F} → G^{∧} → W_{F}

is the identity. Each splitting defines a homomorphism of η of W_{F} into the group of automorphism of G^{∧}. The splitting will be called admissible if, for each ω in W_{F}, η(ω) is complex analytic and the associated linear transformation of the Lie algebra of Gô is semisimple. It will be called distinguished if there is a Bô, a T ô, and a collection {X_{α}^{∧}} such that η factors through a homomorphism of 𝔤(F̄/F) into A(Gô, Bô, T ô, X_{^{∧}α})). Two distinguished splittings will be called equivalent if they are conjugate under Gô.

We introduce a category 𝒢^{∧}(F) whose objects are extensions of the above type, with Gô a connected reductive complex algebraic group, together with an equivalence class of distinguished splittings. These we call special. A homomorphism

φ : G1^{∧} → G2^{∧}

of two objects in the category will be called an L-homomorphism if

G1^{∧}  ⟶^{φ} G2^{∧}
↘  ↙
W_{F}

is commutative, if the restriction of φ to Gô ; 1 is complex analytic, and if φ preserves admissible splittings. Two L-homomorphisms will be called equivalent if there is a 𝔤 ∈ Gô ; 2 such that

φ_{2} = adg ∘ φ_{1}.

An arrow in our category will be an equivalence class of L-homomorphisms. For simplicity, we do not distinguish in the notation between a homomorphism and its equivalence class.

For future reference we define a parabolic subgroup P^{∧} of G^{∧} to be a closed subgroup P^{∧} such that P ô = P^{∧} ∩ Gô is a parabolic subgroup of Gô and such that the projection P^{∧} → W_{F} is surjective.

We also remark that A(Gô, Bô, T ô, {X_{α^{∧}}}) contains no inner automorphisms. Thus if

η : 𝔤(\overline{F} /F) → A(Gô, Bô, T ô, \{X_{α^{∧}}\}),
\overline{η} : 𝔤(\overline{F} /F) → A(Gô, \overline{B}ô, \overline{T} ô, \{\overline{X}\overline{α}^{∧}\})

are associated to two distinguished splittings of G^{∧} there is a g ∈ Gô, unique modulo the center, such that

\overline{η} = adg ∘ η ∘ adg^{−1}.

Suppose we are given a special distinguished splitting associated to the above map η. Let L^{∧} be the group of rational characters of T ô. If both variables on the right are treated as algebraic groups

L^{∧} = Hom(T ô, C^{×}).

Let conversely

L = Hom(C^{×}, T ô).

Define a pairing

L × L^{∧} → Z

by

λ^{∧}(λ(z)) = z^{(λ,λ^{∧})}, z ∈ C^{×}.

This pairing identifies L^{∧} with Hom(L, Z). Associated to each root α^{∧} of T ô is a homomorphism of SL(2, C) into Gô. The composition

z → ( \begin{matrix}z & 0 \\ 0 & z^{−1} \end{matrix} ) → Gô

factors through T ô and defines an element α of L.

Let Δ^{∧} be the set of roots simple with respect to Bô. Associated to Gô, Bô, T ô, {X_{α^{∧}}} are a connected reductive group Gô over F, a Borel subgroup Bô of Gô, a Cartan subgroup T ô in Bô, and isomorphisms η_{α}, α^{∧} ∈ Δ^{∧}, of the additive group with a subgroup of Bô such that

L = Hom(T, GL(1))

and

Δ = {α|α^{∧} ∈ Δ^{∧}}

is the set of simple roots of T ô with respect to Bô. Moreover

adt(η_{α}(x)) = η_{α}(α(t)x), x ∈ \overline{F} , t ∈ T ô(\overline{F}).

The collection Gô, Bô, T ô, η_{α} is determined up to canonical isomorphism by these conditions. Any ω in A(Gô, Bô, T ô, {X_{α^{∧}}}) acts on L and L^{∧}. There is a unique way of letting ω act on Gô so that

ωλ(ωt) = λ(t), λ ∈ L, t ∈ T ô(\overline{F}),

and

ωη_{α}(x) = η_{ωα}(x), x ∈ \overline{F} .

The automorphism ω so obtained is defined over F. Thus

η : 𝔤(\overline{F} /F) → A(Gô, Bô, T ô, \{X_{α^{∧}}\})

defines an element of H^{1}(𝔤(\overline{F} /F), AutG^{o}) and hence an F -form G of G^{o}. In particular

G(F) = \{g ∈ G^{o}(\overline{F})|τη(τ)(g) = g)∀τ ∈ 𝔤(\overline{F} /F)\}.

Observe that the group G is quasi-split. Observe also that the data associated to two special distinguished splittings of G^{∧} are connected by a unique inner automorphism. It follows readily that the group G, together with B, T, {η_{α}}, is determined up to canonical isomorphism by G^{∧}.

Conversely, suppose we are given a quasi-split group G over F. Choose a Borel subgroup B and a Cartan subgroup T in B all defined over F. Interchanging the roles of L and L^{∧} and of Δ and Δ^{∧}, we pass from G, B, and T to Gô, Bô, T ô, and {X_{α}}. The group A(Gô, Bô, T ô, {X_{α^{∧}}}) may be identified with the group of automorphisms of L that leave the set Δ invariant. Define a homomorphism

η : 𝔤(\overline{F} /F) → A(Gô, Bô, T ô, \{X_{α^{∧}}\})

by

η(τ)λ(τ(t)) = τ(λ(t)), λ ∈ L, t ∈ T (\overline{F}).

This map allow us to define G^{∧}, which again is determined up to canonical isomorphisms by G alone.

Suppose G_{1} and G_{2} are two quasi-split groups over F and ψ : G_{1} → G_{2} is an isomorphism with ψ^{−1}τ(ψ) inner for each τ in 𝔤(\overline{F} /F). Choose 𝔤 ∈ G_{1}(\overline{F}) so that ψ′ = ψ ∘ adg takes B_{1} to B_{2} and T_{1} to T_{2}. Then ψ′ determines a bijection Δ_{1} → Δ_{2} as well as an isomorphism ψ′ : L_{1} → L_{2}. These do not depend on the choice of g and determine an isomorphism ψ^{∧} : Gô ; 1 → Gô ; 2. This isomorphism takes Bô ; 1 to Bô ; 2, T ô ; 1 to T ô ; 2, and X_{α_{1}^{∧}} to X_{α_{2}^{∧}} if α_{1} and α_{2} are corresponding elements in Δ_{1} and Δ_{2}. Since ψ^{′−1}τ(ψ′) takes T_{1} to T_{1}, B_{1} to B_{1}, and is inner it is the identity on T_{1}. It follows readily that

η_{2}(τ)ψ′(λ_{1}) = ψ′(η_{1}(τ)λ_{1}).

Thus ψ^{∧} may be extended to an isomorphism of G1^{∧} with G2^{∧} that preserves the splittings. It is determined uniquely by the conditions imposed upon it.

These are of course the considerations which allowed us to define G^{∧} in the first place. If G_{1} = G_{2} = G then G^{∧} may be realized either as Gô ; 1 × W_{F} or as Gô ; 2 × W_{F} but these two groups are canonically isomorphic. There are occasions when a failure to distinguish between G and its realizations leads to serious confusion.

In general if G_{1} is a connected reductive group over F we may choose an isomorphism ψ of G_{1} with a quasi-split group G. ψ is to be defined over \overline{F} and ψ^{−1}τ(ψ) is to be inner for τ ∈ 𝔤(\overline{F} /F). We may, taking into account the canonical isomorphisms above, define G1^{∧} to be G^{∧}. However, we should observe that the same difficulties are present here as in the definition of the fundamental group; the isomorphism ψ we write

G1^{∧}(ψ).

There are some further observations to be made before the task of this paper can be formulated. Let 𝔭(G) and 𝔭(G_{1}) be respectively the sets of conjugacy classes of parabolic subgroups of G and G_{1} that are defined over F. Let 𝔭(G^{∧}) be the classes of parabolic subgroups of G^{∧} with respect to conjugacy under Gô. We want to describe a bijection

𝔭(G) ↔ 𝔭(G′)

and an injection

𝔭(G_{1}) ↪ 𝔭(G^{∧}).

For the first we recall that for a given T and B and the corresponding T ô, Bô we have a bijection Δ ↔ Δ^{∧}. It is well known that 𝔭(G) is parametrized by the 𝔤(\overline{F} /F)-invariant subsets of Δ. The classes of parabolic subgroups of Gô are parametrized by the subset of Δ. The normalizer of P ô in G^{∧} is parabolic if and only if the associated subset of Δ is invariant under 𝔤(\overline{F} /F). This yields the bijection.

The injection will now be defined by

𝔭(G_{1}) ↪ 𝔭(G).

Suppose P_{1} is a parabolic subgroup of G_{1} defined over F. I claim that here there is a g in G_{1}(\overline{F}) such that if ψ′ = ψ ∘ adg then P = ψ′(P_{1}) is defined over F. The class of P depends only on ψ and the class of P_{1}. The required injection maps the latter class to the former. To prove that g exists we use the following lemma.

LEMMA 2.1. Let G′ and G be connected reductive groups over F. Let G be quasi-split and let ψ : G′ → G be an isomorphism defined over \overline{F}. Suppose ψ^{−1}τ(ψ) is inner for τ ∈ 𝔤(\overline{F} /F). If T′ is a Cartan subgroup of G′ defined over F there is a g′ ∈ G′(\overline{F}) and a Cartan subgroup T in G defined over F such that ψ′ = ψ ∘adg′ when restricted to T′ yields an isomorphism of T′ with T that is defined over F.

Let G′_{der} be the derived group of G′ and let G′_{sc} be its simply connected covering group. Define G_{der} and G_{sc} in the same way. Lift ψ to an isomorphism ψ_{sc} : G′_{sc} → G_{sc}. Let T_{sc} be the inverse image of T′ in G′_{sc}. Choose t̃′ ∈ T′_{sc}(F) with image t′ in T′(F) so that T′_{sc} is the centralizer of t′ and T′ the centralizer of t′. Set t̃_{1} = ψ_{sc}(t′). Since

τ(t̃_{1}) = ψ_{sc}(ψ_{sc}^{−1}τ(ψ_{sc})(t̃′)), τ ∈ 𝔤(\overline{F} /F),

the conjugacy class of t̃_{1} is defined over F. By Theorem 1.7 of [14] there is a g̃ ∈ G_{sc}(\overline{F}) such that t̃= adg̃(t̃_{1}) lies in G_{sc}(F). Let t be its projection in G(F). The centralizer T of t is defined over F and if g′ is the projection of g̃′ = ψ_{sc}^{−1}(g̃) then ψ′ = ψ ∘ adg maps t′ to t and T′ to T. Since both t′ and t are rational over F the automorphism ψ^{′−1}τ(ψ′) which is inner commutes with t′ and hence with all of T′. It follows that ψ′ : T′ → T is defined over F.

We apply the lemma with G′ equal to G_{1} and with T′ equal to a Cartan subgroup T_{1} lying in P_{1}. Choose g so that if ψ′ = ψ ∘ adg then ψ^{′−1}τ(ψ′) lies in T_{1}(\overline{F}) for τ ∈ 𝔤(\overline{F} /F). Then if

P = ψ′(P_{1})

we have

τ(P) = ψ′(ψ^{′−1}τ(ψ′)(P_{1})) = ψ(P_{1}) = P

and P is defined over F.

Let 𝔭^{∧}(G_{1}) be the image of 𝔭(G_{1}) in 𝔭(G1^{∧}).

LEMMA 2.2. If \overline{P} 1^{∧} ⊇ P1^{∧} and the class of P1^{∧} lies in 𝔭^{∧}(G_{1}) so does the class of \overline{P} 1^{∧}.

Choose P_{1} in G_{1} that is defined over F. The parabolic subgroups of G_{1} that are defined over F and contain P_{1} belong to different classes. So do the parabolic subgroups of G1^{∧} that contain P1^{∧}. We have only to verify that these sets contain the same number of elements. Choose T_{1} in P_{1} that is defined over F and choose an isomorphism ψ of G_{1} with a quasi-split group G so that ψ^{−1}τ(ψ) is inner and commutes with T_{1} for all τ ∈ 𝔤(\overline{F} /F). Let M_{1} be a Levi factor of P_{1} containing T_{1} and let S_{1} be a maximal torus in the center of M_{1}. Then P = ψ(P_{1}), M = ψ(M_{1}), and S = ψ(S_{1}), as well as ψ|S_{1} are all defined over F. Thus the maximal split tori in S and S_{1} have a common rank r and P and P_{1} are both contained in 2^{r} parabolic subgroups defined over F. Since the number of parabolic subgroups of G1^{∧} that contain P1^{∧} is equal to the number of parabolic subgroups of G that are defined over F and contain P the required equality follows.

The group W_{F} lies in 𝒢^{∧}(F). Let Φ(G_{1}) be the subset of

Hom_{𝒢^{∧}(F)} (W_{F} , G1^{∧})

consisting of these φ such that the class of any parabolic subgroup P^{∧} containing φ(W_{F}) lies in 𝔭^{∧}(G_{1}) under the above injection. In particular, for the quasi-split group G

Φ(G) = Hom_{𝒢^{∧}(F)} (W_{F} , G^{∧})

which is obviously a covariant functor of G^{∧}.

We shall start in the next paragraph to relate Φ(G) to Π(G). There are some simple properties of Φ(G) to establish first. The group G(F) does not change on restriction of scalars and neither does Π(G). We had best check that this is also true for Φ(G). Although there is, in the present circumstances, only one nontrivial way to restrict scalars, namely from C to R, I would prefer not to take this explicitly into account.

Let E be a finite extension of F. We want first of all to define a faithful functor from 𝒢^{∧}(E) to 𝒢^{∧}(F). We imbed E in \overline{F}. Corresponding to this imbedding is an imbedding of W_{E} in W_{F}. Actually there is some arbitrariness in both imbeddings. Since, up to equivalence, it has no effect on the functor to be constructed, we ignore it. Suppose G̃^{∧} lies in 𝒢^{∧}(E). Choose a distinguished splitting of G̃^{∧} and let η̃ be the corresponding action of W_{E} on G̃ô. Let Gô be the set of functions h on W_{F} with values in G̃ô satisfying

h(vw) = η̃(v)(h(w)), v ∈ W_{E}.

Let η(v), v ∈ W_{F}, send h to h′ with

h′(w) = h(wv).

With respect to this action form the semidirect product

G^{∧} = Gô × W_{F} .

It is easy to see that the given splitting of G^{∧} is distinguished and that G^{∧} lies in 𝒢^{∧}(F). Observe also that there is an obvious bijection from 𝔭(G̃^{∧}) to 𝔭(G^{∧}).

If we had chosen another distinguished splitting η̃′ there would be a g ∈ G̃ô such that

η̃′(w) = gη̃(w)g^{−1}, w ∈ W_{E}.

The map h → h′ with h′(w) = gh(w)g^{−1} together with the identity on W_{F} would yield an isomorphism between G^{∧} and the group constructed by means of η̃′; so we need not worry about the arbitrariness of the distinguished splitting either.

Choose a set V of representatives v for W_{E}\W_{F}. If w ∈ W_{F} let

vw = d_{v}(w)v′, v, v′ ∈ V.

If φ̃ is an L-homomorphism from G̃1^{∧} to G̃2^{∧} let

φ̃(1 × w) = ã(w) × w, w ∈ W_{E},

with respect to special distinguished splittings of G̃1^{∧} and G̃2^{∧}. If w ∈ W_{F} let a(w) be the function in Gô ; 2 whose value at v ∈ V is ã(d_{v}(w)). If h is a function in Gô ; 1 let h′ be the function in Gô ; 2 defined by

h′(v) = φ̃(h(v)), v ∈ V.

Define G1^{∧} and G2^{∧} as above and let φ be the homomorphism from the former to the latter defined by

φ(h × w) = h′a(w) × w.

A little calculation that will be left to the reader shows that φ is in fact an L-homomorphism and that its class is determined by that of φ̃ alone and is independent of the auxiliary data. The reader will also easily verify that the class determined by φ̃_{1}φ̃_{2} is φ_{1}φ_{2}.

Given φ we define φ̃ as follows. If h̃ × w, w ∈ W_{E}, belongs to G̃1^{∧} we let h be a function in Gô ; 1 with h(1) = h̃. If

φ(h × w) = h′ × w

we set

φ̃(h̃ × w) = h′(1) × w.

The class of φ̃ depends only on that of φ. It is clear that this process inverts the operation of the previous section. A slight variant of Shapiro’s lemma shows that the reciprocal is true. Starting from φ we construct φ̃; from φ̃ we pass to φ′. We have to show that φ and φ′ lie in the same class. We may assume that the set of representatives V contains 1. Suppose

φ(w) = h_{w} × w

and define h in Gô ; 2 by

h(v) = h_{v}(1).

It is easily verified that

φ(g) = hφ′(g)h^{−1}, g ∈ G1^{∧}.

Thus our functor is fully faithful. The object of 𝒢^{∧}(F) corresponding to W_{E} is W_{F}.

Suppose G̃ is quasi-split over E and G over F is obtained from G̃ by restriction of scalars. Then for any scheme Z over F

Hom_{F} (Z, G) = Hom_{E}(Z ⊗_{F} E, G̃)

because restrictions of scalars is the right adjoint of base change. In particular if a Borel subgroup B̃ of G̃ and a Cartan subgroup T̃ of B̃ are given, then restriction of scalars yields B and T in G; so G is quasi-split. We must verify that G^{∧} is obtained from G̃^{∧} by the functor introduced above.

Let L′ be the group of functions λ′ on 𝔤(\overline{F} /F) with values in L̃^{∧} satisfying

λ′(στ) = σλ′(τ), σ ∈ 𝔤(\overline{F} /E),

and let Δ′ be the set of λ′ that are zero on all but one coset of 𝔤(\overline{F} /E) on which they take values in Δ̃. All we have to do is show that L′ is isomorphic to L^{∧} as a 𝔤(\overline{F} /F) module in such a way that Δ′ corresponds to Δ.

Since we have chosen an imbedding of E in F we may take \overline{E} to be \overline{F}. Map \overline{F} ⊗_{F} E to the ring R of \overline{F} -valued functions a on 𝔤(\overline{F} /F) satisfying

a(στ) = σ(a(τ)), σ ∈ 𝔤(\overline{F} /E),

by

α ⊗ β → a : τ → τ(α)β.

This is an isomorphism. Then

L^{∧} = Hom\overline{F} (GL(1) ⊗_{F} \overline{F} , T ⊗_{F} \overline{F})
= Hom_{F} (GL(1)⊗_{F} , \overline{F} , T) = Hom_{E}(GL(1) ⊗_{F} R, T̃).

Every τ ∈ 𝔤(\overline{F} /F) yields by evaluation a map R → \overline{F} and hence a map

Hom_{E}(GL(1) ⊗_{F} R, T̃) → Hom_{E}(GL(1) ⊗_{F} \overline{F} , T̃) = L̃^{∧}.

Thus every element of L^{∧} yields a function on 𝔤(\overline{F} /F) with values in L̃^{∧}. The function is easily seen to lie in L′. That the resulting map from L^{∧} to L′ has the required properties is easy to see.

If we take L to be Hom(L, Z) we may identify L with the space of functions λ on 𝔤(\overline{F} /F) with values in L̃ satisfying

λ(στ) = σ(λ(τ)), σ ∈ 𝔤(\overline{F} /E).

The pairing is

⟨λ, λ^{∧}⟩ = ∑_{𝔤(\overline{F} /E)\𝔤(\overline{F} /F)}⟨λ(τ), λ^{∧}(τ)⟩.

If z is an \overline{F} -valued point in GL(1) then

λ(λ^{∧}(z)) = z^{(λ,λ^{∧})} = z^{∑<λ(z),λ^{∧}(τ)>} = ∏ λ(τ)(λ^{∧}(τ)(z)) = ∏ τ^{−1}{λ(τ)(λ^{∧}(τ)(τz))}

because every rational character of GL(1) is defined over F. In general we have an isomorphism

T (\overline{F}) = Hom_{F} (Spec \overline{F} , T) = Hom_{E}(Spec R, T̃) = T̃(R).

Since each τ ∈ 𝔤 (\overline{F} /F) yields a map R → \overline{F}, we may associate to each s ∈ T (\overline{F}) a function τ → s(τ) on 𝔤(\overline{F} /F) with vaules in T̃(\overline{F}). If s = λ^{∧}(z) then s(τ) = λ^{∧}(τ)(τ(z)). Since the points λ^{∧}(z) generate T (\overline{F}) we have

λ(s) = ∏ τ^{−1}{λ(τ)(s(τ))}.

In particular if s lies in T (F) then s(τ) = s̃ is independent of τ and lies in T̃(E).

It has already been pointed out that the definition of the associated group of an arbitrary connected reductive group G_{1} depends on the choice of an isomorphism ψ : G_{1} → G with G quasi-split. However, composing ψ with an inner automorphism has no effect on the construction. In particular, since ψ^{−1}τ(ψ), τ ∈ 𝔤(\overline{F} /F) is always supposed inner, ψ could be replaced by τ(ψ).

LEMMA 2.3. Suppose G̃_{1} and G̃ are given over E with G̃ quasi-split, together with an isomorphism ψ̃ : G̃_{1} → G̃ over \overline{E}. Let G_{1} and G be obtained from G̃_{1} and G̃ by restriction of scalars. There is associated to ψ̃ an isomorphism ψ : G_{1} → G over \overline{F} defined up to composition with an inner automorphism and G1^{∧} is obtained from G̃1^{∧}(ψ̃) by the restriction of scalars functor from 𝒢^{∧}(E) to 𝒢^{∧}(F).

Only the existence of ψ needs to be established. We imbed E in F and identify \overline{E} with \overline{F}

Hom_{F} (G_{1} ⊗_{F} \overline{F} , G ⊗_{F} \overline{F}) = Hom_{F} (G_{1} ⊗_{F} \overline{F} , G)
= Hom_{E}(G_{1} ⊗_{E} (G_{1} ⊗_{F} R, G̃)

and

Hom_{F} (G_{1} ⊗_{F} \overline{F} , G_{1} ⊗_{F} \overline{F}) = Hom_{E}(G_{1} ⊗_{F} R, G̃_{1}).

Start from the identity morphism on the left to get a morphism from G_{1} ⊗_{F} R to G̃_{1}. On the other hand, if we choose a set of representatives ρ for 𝔤(\overline{F} /E) in 𝔤(\overline{F} /F) we may imbed \overline{F} in R by associating to α ∈ \overline{F} the function whose value at each ρ is α. This yields a morphism from Spec R to Spec \overline{F} over E. The two morphisms together yield a morphism from G_{1} ⊗_{F} R to G̃_{1} ⊗_{E} \overline{F}. Composing with ψ̃ : G̃_{1} ⊗_{E} \overline{F} → G̃ we get a morphism from G_{1} ⊗_{F} R to G̃ and hence ψ : G_{1} ⊗_{F} \overline{F} → G.

The invariance of Φ(G) under restriction of scalars is now clear. Suppose P is a parabolic subgroup of G over F. We may choose B and T in P. Now construct G^{∧}, Bô, and T ô. Let P^{∧} be the parabolic subgroup of G^{∧} containing Bô whose class corresponds to that of P. Let N be the unipotent radical of P, N^{∧} that of P ô, and let M = P/N, M^{∧} = P^{∧}/N^{∧}. It is easily seen that M^{∧} belongs to 𝒢^{∧}(F) and that M^{∧} is the associated group of M. If \overline{P}^{∧} is another parabolic subgroup in the same class as P^{∧} there is a g ∈ Gô such that gP^{∧}g^{−1} = \overline{P}^{∧}. The induced map M^{∧} → \overline{M}^{∧} is uniquely determined up to an inner automorphism by an element of \overline{M} ô. Thus if \overline{P}^{∧} and P lie in corresponding classes in 𝔭(G^{∧}) and 𝔭(G) the associated group of M is canonically isomorphic, in the category 𝒢^{∧}(F), \overline{P}^{∧}/\overline{N}^{∧}.

Suppose ψ : G_{1} ⟶^{∼} G is such that ψ^{−1}τ(ψ) is inner for τ ∈ 𝔤(\overline{F} /F). If P_{1} is a parabolic subgroup of G_{1} over F we may always modify ψ by an inner automorphism so that P = ψ(P_{1}) is defined over F. We readily deduce the following lemma.

LEMMA 2.4. Suppose P_{1} is a parabolic subgroup of G_{1} over F and P1^{∧} is a parabolic subgroup of G1^{∧} whose class corresponds to that of P_{1}. Then M1^{∧} = P1^{∧}/N1^{∧} is canonically isomorphic in the category 𝒢^{∧}(F) to the associate group of M_{1}.

Choose a splitting M_{1} → P_{1} defined over F and a splitting M1^{∧} → P1^{∧} that carries distinguished splittings of M1^{∧} to distinguished splittings of G1^{∧}. The isomorphism between M1^{∧} and the associated group of M_{1} depends on the choice of P_{1} and P1^{∧} with M_{1} and M1^{∧} as Levi factors.

LEMMA 2.5. Suppose P_{1} and P1^{∧} are given as above with M_{1} and M1^{∧} as Levi factors. There is a bijection η between the parabolic subgroups of G_{1} defined over F that contain M_{1} as a Levi factor and the parabolic subgroups of G1^{∧} that contain M1^{∧} as Levi factor such that π1^{∧} = η(P_{1}), and such that the isomorphism between M1^{∧} and the associated group of M_{1} is the same for all pairs P1^{∧}, η(\overline{P}_{1}).

Take G quasi-split and let ψ be an isomorphism from G_{1} to G with ψ^{−1}τ(ψ) inner for τ ∈ 𝔤(\overline{F} /F). We also suppose that there is a Cartan subgroup T_{1} in M_{1} defined over F such that each ψ^{−1}τ(ψ) commutes with the elements of T. Then ψ(T_{1}), M = ψ(M_{1}), and P = ψ(P_{1}) are defined over F. In fact if \overline{P}_{1} is any parabolic subgroup over F that contains M_{1} then \overline{P} = ψ(\overline{P}_{1}) is defined over F. The definitions are such that we may prove the assertions for G, M, P rather than G_{1}, M_{1}, P_{1}. Choose a Borel subgroup B over F that is contained in P and a Cartan subgroup T of B that is also defined over F. Then build G^{∧}, Bô, T ô, and {X_{α^{∧}}}. We may replace G1^{∧} by G^{∧} and suppose that P^{∧} contains B^{∧}. Since any two Levi factors of P^{∧} are conjugate under P ô (cf. [12], Theorem 7.1), we may also suppose that M^{∧} contains T^{∧}.

Let D(M) be the space of vectors in L ⊗ R invariant under 𝔤(\overline{F} /F) and orthogonal to the roots of M^{∧}. By a chamber in D(M) we mean a connected component of the complement of the union of the hyperplanes

{a ∈ D(M)|⟨a, α^{∧}⟩ = 0}

where α^{∧} is a root of T ô in Gô but not in M ô. There is a bijection between chambers in D(M) and parabolic subgroups \overline{P}^{∧} of G^{∧} that contain M^{∧} as Levi factor. \overline{P}^{∧} corresponds to the chamber

C = {a ∈ D(M)|⟨a, α^{∧}⟩ > 0 ifX_{α^{∧}} ∈ 𝔭^{∧}, X_{α^{∧}} ∉ m^{∧}}.

𝔭 and 𝔪^{∧} are the Lie algebras of P^{∧} and M^{∧}.

There is also a bijection between chambers of D(M) and parabolic subgroups of G that are defined over F and contain M as Levi factor. If B is the Killing form, which may be degenerate, then C corresponds to \overline{P} defined by the condition that it contain T and that a root α of T in G be a root in \overline{P} if and only if B(a, α) ≥ 0 for all a in C. The bijection η is the composition of \overline{P} → C → \overline{P}^{∧}.

The Weyl groups Ω^{∧} and Ω of T ô in Gô and of T in G are isomorphic in such a way that the reflections

λ → λ − ⟨λ, α^{∧}⟩α,
λ^{∧} → λ^{∧} − ⟨α, λ^{∧}⟩α^{∧}

correspond. Suppose \overline{P}^{∧} = η(\overline{P}). There is an ω^{∧} in Ω^{∧} that takes every root of T ô in \overline{P} ô not in M ô and every root in M ô ∩ Bô to a root of T ô in Bô. Let h in the normalizer of T ô in Gô represent ω^{∧} and let P0^{∧} be h\overline{P}^{∧}h^{−1}, and M0^{∧} be hM^{∧}h^{−1}. We may suppose that

Adh(X_{α^{∧}}) = X_{ω^{∧}(α^{∧})}

if α^{∧} is a root of T ô in M ô ∩ Bô. If g in the normalizer of T in G(\overline{F}) represents the element ω of Ω corresponding to ω^{∧} then P_{0} = g\overline{P} g^{−1} contains B. It is clear that α is a root of T in P_{0} if and only if α^{∧} is a root of T ô in P0^{∧}. Thus P_{0} and P0^{∧} and hence \overline{P} and \overline{P}^{∧} belong to corresponding classes in 𝔭(G) and 𝔭(G^{∧}).

If we build the associate group of M starting with M, B ∩ M, and T we obtain M^{∧}, B^{∧} ∩ M^{∧}, T ô, and the collection {X_{α^{∧}}} where α^{∧} runs over the simple roots of T ô in M ô with respect to Bô ∩ M ô. This gives the isomorphism of M^{∧} with the associate group of M defined by P and P^{∧}. The isomorphism between M^{∧} and the associated group of M defined by \overline{P} and \overline{P}^{∧} is more complicated to obtain. This is not because of any intrinsic asymmetry but rather because of the simplifying assumption that P^{∧} contains B^{∧} and P contains B. We have to use g to establish an isomorphism between M and M_{0} = gMg^{−1} that we may assume is defined over F, then build the associate group of M_{0} with respect to B ∩ M_{0} and T, obtaining thereby M0^{∧}, B^{∧} ∩ M0^{∧}, T ô, and {X_{α^{∧}}}, where α^{∧} runs over the simple roots of T ô in M ô with respect to Bô ∩ M ô ; 0, and finally we have to use the isomorphism between M^{∧} and M0^{∧} given by h.

What has to be verified to prove the lemma is that, in the category 𝒢^{∧}(F), the isomorphism between M^{∧} and M0^{∧} given by h is equal to the isomorphism between them as two concrete realizations of the associate group of M. What is the latter isomorphism? The isomorphism adg takes M to M_{0}, B ∩ M to B ∩ M_{0}, T to T, and the root α of T in M to ωα. Then the isomorphism between M^{∧} and M0^{∧} as realizations of the associate group takes M ô to M ô ; 0, Bô ∩ M ô to Bô ∩ M ô ; 0, T ô to T ô, X_{α^{∧}} to X_{ω^{∧}α^{∧}}, respects the splittings M^{∧} = M ô × W_{F}, M0^{∧} = M ô ; 0 × W_{F} built into the construction, and acts trivially on W_{F}. It is characterized by these properties. Since (ωα)^{∧} = ω^{∧}α^{∧} the isomorphism given by h has all these properties except perhaps the last. To achieve the last we exploit the circumstance that we are not really working with isomorphisms but rather with classes of them to modify our initial choice of h.

W_{F} acts on L^{∧}. Since in its action on Gô it leaves \overline{P} ô, M ô, and Bô invariant and since the normalizer of T ô in Bô is T ô, it is clear that on L^{∧}

wω^{∧} = ω^{∧}w, w ∈ W_{F} .

That h can be modified in the fashion desired follows immediately from the next lemma.

LEMMA 2.6. Let G^{∧}, Bô, T ô, and {X_{α^{∧}}} be given. Suppose ω^{∧} ∈ Ω^{∧} and that on L^{∧}

wω^{∧} = ω^{∧}w, w ∈ W_{F} .

Then ω^{∧} is represented by an element h of the normalizer of T ô in Gô that commutes with w in W_{F} and satisfies

adh(X_{α^{∧}}) = X_{ω^{∧}α^{∧}}

if α^{∧} is simple with respect to Bô.

We ignore for the moment the last condition and simply try to find an h that represents ω^{∧} and it is fixed by the action of W_{F} on Gô. The action of W_{F} on Gô factors through 𝔤(\overline{F} /F) and it is easier to forget about W_{F} and deal directly with 𝔤(\overline{F} /F). Start off with any h that represents ω^{∧}. Then

τ → a_{τ} (h) = τ(h)h^{−1}

lies in T ô and is a 1-cocycle of 𝔤(\overline{F} /F) with values in T ô. If h is replaced by sh, s ∈ T ô, then a_{τ} (h) is replaced by τ(s)a_{τ} (h)s^{−1}; so our problem is to show that the class of the cocycle is trivial. Since

a_{τ} (h_{1}h_{2}) = a_{τ} (h_{1})ω1^{∧}(a_{τ} (h_{2}))

it will be enough to show this for a set of generators of the centralizer Ω0^{∧} of 𝔤(\overline{F} /F) in Ω^{∧}.

Suppose A is the set of vectors in L ⊗ R invariant under 𝔤(\overline{F} /F). The group Ω0^{∧} acts faithfully on A and, as is easily seen, acts simply transitively on the chambers, that is, the connected components of the complement of the hyperplanes.

{a ∈ A|⟨a, α^{∧}⟩ = 0}

where α^{∧} is any root of T ô in Gô. Each orbit O in Δ^{∧} defines a reflection

S_{O} : a → a − ⟨a, α0^{∧}⟩ ∑_{α^{∧}∈O} α

where α0^{∧} is any element of O. These reflections are each given by an ω0^{∧} in Ω0^{∧} and the collection of ω0^{∧} generates Ω0^{∧}. We have to show that each ω0^{∧} is represented by an element of Gô that is fixed by 𝔤(\overline{F} /F). Replacing Gô by a subgroup if necessary, we may suppose that 0 = Δ^{∧}. Since the question only becomes more difficult if Gô is replaced by a finite covering group, we may suppose Gô is the product of a torus and a finite number of simple, simply connected groups. The torus may be discarded. Let

Gô = ∏_{i=1}^{r} Gô ; i, T ô = ∏_{i=1}^{r} T ô ; i, Ω^{∧} = ∏_{i=1}^{r} Ωi^{∧},

and

ω^{∧} = ωΔ^{∧}^{∧} = ∏_{i=1}^{r} ωi^{∧}.

If τ(Gô ; i) = Gô ; j then

τ(ωi^{∧}) = ωj^{∧}.

Suppose 𝔤(\overline{F} /E) is the stabilizer of Gô ; 1 in 𝔤(\overline{F} /F). Then ω1^{∧} commutes with 𝔤(\overline{F} /F). Suppose it is represented by h_{1} in Gô ; 1 which is fixed by 𝔤(\overline{F} /E). Set

h_{j} = τ(h_{1})

where τ is any element of 𝔤(\overline{F} /F) that takes Gô ; 1 to Gô ; j. Then h_{j} is well defined and

h = ∏_{j=1}^{r} h_{j}

is fixed by 𝔤(\overline{F} /F) and represents ω^{∧}.

We are now reduced to a situation in which Gô is simple and simply connected and 𝔤(\overline{F} /F) acts transitively on Δ^{∧}. There are two possibilities. The group Gô is of type A_{1} or A_{2}. In the first case 𝔤(\overline{F} /F) acts trivially and there is nothing to prove. In the second we may take Gô to be SL(3, C), T ô to be the group of diagonal matrices, Bô to be the group of upper triangular matrices, and the collection {X_{α^{∧}}} to consist of

( \begin{matrix}0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) ,  ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix} ) .

Then A(Gô, Bô, T ô, {X_{α^{∧}}}) consists of the trivial automorphism and the automorphism

H → ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} )^{t}H^{−1} ( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} ) .

We may take h to be

( \begin{matrix}0 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & 0 & 0 \end{matrix} ) .

Suppose ω^{∧} is arbitrary in Ω_{0}^{∧} and is represented by an h in Gô that is fixed by 𝔤(\overline{F} /F). In order to complete the proof of the lemma we have to show that there is an s in T ô that is fixed by 𝔤(\overline{F} /F) such that

ad(hs)X_{α^{∧}} = X_{ω^{∧}α^{∧}}, α^{∧} ∈ Δ^{∧}.

Let

adh(X_{α^{∧}}) = c(α^{∧})X_{ω^{∧}α^{∧}}.

Clearly c(τα^{∧}) = c(α^{∧}) for τ ∈ 𝔤(\overline{F} /F). We may choose d(α^{∧}), α^{∧} ∈ Δ^{∧}, such that d(τα^{∧}) = d(α^{∧}) and such that

d(α^{∧})^{|𝔤(\overline{F} /F)|} = c(α^{∧}).

If t in T ô satisfies

α^{∧}(t) = d(α^{∧})^{−1}, α^{∧} ∈ Δ^{∧},

then

s = ∏_{τ∈𝔤(\overline{F} /F)} τ(t)

is the required s.

Suppose φ_{1} is an automorphism of G_{1} such that φ_{1}^{−1}τ(φ_{1}) is inner for all τ ∈ 𝔤(\overline{F} /F). For example φ_{1} could be defined over F. If ψ is an isomorphism of G_{1} with a quasi-split group G, we define the automorphism φ of G by transport of structure. We have seen already that φ determines an automorphism φ^{∧} of G^{∧}. By transport of structure again we obtain an automorphism φ1^{∧} of G1^{∧}. It is easily seen that φ1^{∧} depends only on φ_{1} and not on ψ.

LEMMA 2.7. Suppose P_{1} is a parabolic subgroup of G_{1} over F and P1^{∧} is a parabolic subgroup of G^{∧} whose class corresponds to that of P_{1}. Let M_{1} be a Levi factor of P_{1} over F and M1^{∧}, which we take as the associate group of M_{1}, a Levi factor of P1^{∧}. Suppose g ∈ G_{1}(F) normalizes M_{1}. If φ_{1} is the restriction of Adg to M_{1} and φ1^{∧} the associated automorphism of M1^{∧}, there is an element h in the normalizer of M1^{∧} in Gô ; 1 such that φ_{1}^{∧} is the restriction of Adh to M1^{∧}.

Suppose that g is only in G_{1}(\overline{F}) but that g^{−1}τ(g) lies in M_{1}(\overline{F}) for each τ. Then we can still define φ1^{∧} and the lemma remains valid. We work with the weaker assumption. The advantage is that if ψ is an isomorphism of G_{1} with a quasi-split group G such that ψ^{−1}τ(ψ) = adm_{τ} with m_{τ} ∈ M(\overline{F}) for each τ then ψ(g) continues to satisfy the weaker assumption, for

ψ(g^{−1})τ(ψ(g)) = ψ(g^{−1}m_{τ} τ(g)m_{τ}^{−1}) ∈ M(\overline{F})

if M = ψ(M_{1}). We prove the lemma for the group G. P_{1} is replaced by P = ψ(P_{1}) and M_{1} by M. g is now in G(\overline{F}). We choose B and T such that B ⊆ P and T ⊆ M.

We may compose g with any element of M(\overline{F}) and thus suppose that g(B ∩M)g^{−1} = b∩M, gT g^{−1} = T. Since g is determined by these conditions modulo T,

gτ(g^{−1}) ∈ T, τ ∈ 𝔤(\overline{F} /F).

In particular g represents an element ω of Ω fixed by 𝔤(\overline{F} /F). Let ω^{∧} be the corresponding element of Ω0^{∧}.

We construct G^{∧}, Bô, T ô, and {X_{α^{∧}}} corresponding to G, B, and T and realize ω^{∧} by an h that satisfies the conditions of the preceding lemma. If we take P^{∧} to contain Bô it is clear that Adh is equal to φ1^{∧} on M^{∧}.

For the next lemma we work in the category of tori over F. Suppose S is such a torus. Then S^{∧} admits by construction a special distinguished splitting. Also L^{∧} is a covariant functor of S and

Sô = Hom(L^{∧}, C^{×})

is a contravariant functor. So is S^{∧}. Φ(S), which consists of classes of continuous homomorphisms of W_{F} into S^{∧}, is also contravariant. We write a homomorphism φ as

φ(w) = a(w) × w.

We compose φ_{1} and φ_{2} by setting

φ_{1}φ_{2}(w) = a_{1}(w)a_{2}(w) × w.

This composition is actually defined for the classes and turns Φ(S) into an abelian group. Π(S) is the group of continuous homomorphisms of S(F) into C^{×}. Although the following lemma is valid over any local field, we prove it here only for the real and the complex field.

LEMMA 2.8. On the category of tori over F the group-valued functors Φ and Π are isomorphic.

When F is C the lemma is particularly easy. Any homomorphism from the topological group C^{×} to C^{×} may be written as

z = e^{x} → z^{a}\overline{z}^{b} = e^{az+b\overline{x}}

where a and b are two uniquely determined elements of C whose difference lies in Z. If φ ∈ Φ(S) is a continuous homomorphism from C^{×} to S^{∧} = Sô, let φ(z) = a(z) × z, z ∈ C^{×}, and

λ^{∧}(a(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}

Classification of irreducible representations 18 where μ and ν are uniquely determined elements of L ⊗ C whose difference lies in L. Associate to φ the element of π of Π(S) defined by

π : t → μ(t)ν(\overline{t}) = e^{⟨μ,H⟩+⟨ν,\overline{H}⟩}

where H ∈ L^{∧} ⊗ C is defined by

λ(t) = e^{⟨λ,H⟩}, λ ∈ L.

That the map φ → π gives the required isomorphism of functors is easily seen.

Now let F be R. Let φ be an honest homomorphism from W_{R} to S^{∧}. Let φ(w) = a(w) × w and λ^{∧}(a(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}, z ∈ C^{×}

If σ is the nontrivial element of 𝔤(C/R) then ν = σμ. Let

a(1 × σ) = α, α ∈ Sô.

and let

λ^{∧}(α) = e^{2π⟨λ_{0},λ^{∧}⟩}, λ_{0} ∈ L ⊗ C.

λ_{0} is determined modulo L and

λ_{0} + σλ_{0} ≡ \frac{1}{2}(μ − ν) (modL).

μ and ν are determined by the class of φ alone but λ_{0} is determined only modulo the sum of L and

{λ − σλ|λ ∈ L ⊗ C}.

We write an element t in S(C) as e^{H} where H in L^{∧} ⊗ C is defined by

λ(t) = e^{⟨λ,H⟩,} λ ∈ L.

t lies in S(R) if and only if

H − σ\overline{H} ∈ 2πiL^{∧}.

Define π by

π(t) = e^{⟨λ_{0},H−σ\overline{H}⟩+⟨μ/2,H+σ\overline{H}⟩}.

This is permissible, for if t is 1 then H ∈ 2πiL^{∧} and

⟨λ_{0}, H − σ\overline{H}⟩ + ⟨\frac{μ}{2}, H + σ\overline{H}⟩ = ⟨λ_{0} + σλ_{0} + \frac{μ}{2} − \frac{σμ}{2}, H⟩ ∈ 2πiZ.

On the other hand, if π is given extend it to a quasi-character π′ of S(C). Let π′(t) = e^{⟨μ_{1},H⟩+⟨μ_{2},\overline{H}⟩.} Define μ and λ_{0} by

μ_{1} = \frac{μ}{2} + λ_{0}, σμ_{2} = \frac{μ}{2} − λ_{0},

so that

μ = μ_{1} + σμ_{2}, λ_{0} = \frac{μ_{1}}{2} − \frac{σμ_{2}}{2} .

Then

λ_{0} + σλ_{0} = \frac{1}{2}\{μ_{1} + σμ_{1} − μ_{2} − σμ_{2}\} ≡ \frac{1}{2}\{μ_{1} + σμ_{2} − σμ_{1} − μ_{2}\}(modL)

and

μ_{1} + σμ_{2} − σμ_{1} − μ_{2} = μ − σμ.

All we have to do is check that μ is determined by π alone and that λ_{0} is determined modulo the sum of L and {λ − σλ|λ ∈ L ⊗ C} by π.

For this we may suppose that π is trivial. If H ∈ L^{∧} ⊗ C then

1 = π′(e^{H+σ\overline{H}}) = e^{⟨μ,H⟩+⟨σμ,\overline{H}⟩}

and μ = 0. If λ^{∧} ∈ L^{∧} and σλ^{∧} = λ^{∧} there is an H ∈ L^{∧} ⊗ C such that

2πiλ^{∧} = H − σ\overline{H}.

Thus

⟨λ_{0}, λ^{∧}⟩ ∈ Z.

It follows immediately that

λ_{0} ∈ L + {λ − σλ|λ ∈ L ⊗ C}.

There is one fact to be verified.

LEMMA 2.9. The functor from Φ to Π respects restriction of scalars.

We consider restriction of scalars from C to R. Let S̃ be a torus over C and S the torus obtained by restriction of scalars. Then

S(R) = Hom_{R}(Spec R, S) ≃ Hom_{C}(Spec C, S̃) = S̃(C).

We denote corresponding elements in S(R) and S̃(C) by s and s̃. L is the group of functions on 𝔤(C/R) with values in L̃ and 𝔤(C/R) operates by right translation. If λ_{1} = λ(1), λ_{2} = λ(σ) then

λ(s) = λ_{1}(s̃)σ(λ_{2}(s̃)).

If s̃ = e^{H^{∼}}, H^{∼} ∈ L̃^{∧} ⊗ C then s = e^{H} with H = (H^{∼}, \overline{H}^{∼}) and

H + σ\overline{H} = 2(H^{∼}, \overline{H}^{∼}), H − σ\overline{H} = 0,

and

e^{⟨μ̃,H^{∼}⟩+⟨ν̃,\overline{H}^{∼}⟩} = e^{⟨λ_{0},H−σ\overline{H}⟩+⟨μ/2,H+σ\overline{H}⟩}

if

μ = (μ̃, ν̃), λ_{0} + \frac{1}{2}(μ̃ − ν̃, 0).

Thus if the quasi-character π̃ of S̃(C) is given by μ̃, ν̃, the associated quasi-character π of S(R) is given by μ and λ_{0}.

One the other hand let φ̃ : W_{C} → S̃^{∧} be given by φ̃(z) = ã(z) × z and let

λ^{∧}(ã(z)) = z^{⟨μ̃,λ^{∧}⟩}\overline{z}^{⟨ν̃,λ^{∧}⟩}.

Sô is the set of functions on 𝔤(C/R) with values in S̃ô. If φ : w → a(w) × w is obtained from φ̃ by the restriction of scalars functor, then

a(z) = (ã(z), ã(\overline{z})), a(1 × σ) = (ã(−1), 1).

One calculates easily the corresponding μ and λ_{0} and finds that they have the correct values.

Now take G connected and reductive. Let Z_{G} be its center. We want to use the previous lemma to associate to each element φ in Φ(G) a homomorphism X_{φ} of Z_{G}(F) into C^{×}. Since Z_{G} is not affected by an inner twisting, we could, but do not, suppose that G is quasi-split. Let G_{rad} be the maximal torus in Z and let G_{ss} be the quotient of G by G_{rad}. If G_{ad} is the adjoint group of G we have the following diagram

Z_{G}

↑ ↘
1 → G_{rad} → G → G_{ss} → 1
↘ ↓
G_{ad}

in which the horizontal line is exact. A pair B, T in G determines B_{ss}, T_{ss} and B_{ad}, T_{ad}. Using these to build the associate groups, we obtain

1 ← Gô ; rad ← Gô ← Gô ; ss ← 1
↖ ↑

Gô ; ad

in which the horizontal line is exact.

In particular we have a map Φ(G) → Φ(G_{rad}), so that every element φ in Φ(G) determines a homomorphism of G_{rad}(F) into C^{×}. Thus when Z_{G} is connected we are able to define X_{φ}. In general let

M = Hom(Z_{G} ⊗ \overline{F} , GL(1)).

M is a 𝔤(\overline{F} /F) module and there is surjection η : L → M whose kernel is the lattice generated by the roots. Let ζ : Q → M be a surjective homomorphism of 𝔤(\overline{F} /F)-modules with Q free over Z. Let

\overline{L} = \{(λ, p)|η(λ) = ζ(p)\}

and let

\overline{Δ} = \{(α, 0)|α ∈ Δ\}.

From \overline{L} and \overline{Δ} and the cocycle defining G we construct \overline{G}. The surjection \overline{L} → L obtained by projection on the first factor yields an injection G → \overline{G} and a surjection \overline{G}^{∧} → G^{∧} whose kernel is a torus over C, namely

Hom(N^{∧}, C^{×}) = Sô

if N is the kernel of \overline{L} → L and S is the torus over F associated to the 𝔤(\overline{F} /F)-module N. Moreover \overline{G}_{rad} = Z\overline{G} is the torus defined by Q.

There is an exact sequence

1 → Hom(S(F), C^{×}) → Hom(Z\overline{G}(F), C^{×}) → Hom(Z_{G}(F), C^{×}).

Every element of Φ(\overline{G}) determines an element of the middle group and hence of the last. If \overline{φ}_{1} and \overline{φ}_{2} in Φ(G) have the same image in Φ(G) then, after an appropriate choice of representatives,

\overline{φ}_{2}(w) = a(w)\overline{φ}_{1}(w)

where a(w) ∈ Sô and

ψ(w) = a(w) × w

is an element of Φ(S). Thus the images of \overline{φ}_{1} and \overline{φ}_{2} in Φ(G_{rad}) differ by an element in the image of Φ(S). By the functoriality of Lemma 2.8, they determine the same element of Hom(Z_{G}(F), C^{×}).

The next lemma will allow us to define X_{φ}, φ ∈ Φ(G); it will remain, however, to verify that it is independent of the choice of Q.

LEMMA 2.10. Suppose \overline{G}^{∧} and G^{∧} are objects in 𝒢^{∧}(F) and φ^{∧} : \overline{G}^{∧} → G^{∧} is a surjective morphism. Suppose that the kernel of φ^{∧} is a torus Sô in the center of \overline{G}ô. Then

Hom(W_{F} , \overline{G}^{∧}) → Hom(W_{F} , G^{∧})

is surjective.

The assumption does not depend on the representative chosen. If T ô is a Cartan subgroup of Gô then

\overline{T} ô = φ^{∧}^{−1}(T ô)

is a Cartan subgroup of \overline{G}ô. If L^{∧} and \overline{L}^{∧} are the lattices of rational characters of T ô and \overline{T} ô then L^{∧} → \overline{L}^{∧} is injective and the quotient is torsion-free. Let

ψ ∈ Hom(W_{F} , G^{∧}).

We may assume that

ψ(C^{×}) ⊆ T ô.

Let

λ^{∧}(ψ(z × 1)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}

where μ and ν lie in L⊗C and μ−ν ∈ L. The map L^{∧} → \overline{L}^{∧} leads to surjective maps \overline{L} → L and \overline{L}⊗C → C. Lift μ to μ̃ and ν to ν̃ in \overline{L} ⊗ C so that μ̃ − ν̃ lies in \overline{L}.

Define ψ̃(z × 1) in \overline{T} ô by

λ^{∧}(ψ̃(z × 1)) = z^{⟨μ̃,λ^{∧}⟩}\overline{z}^{⟨ν̃,λ^{∧}⟩}, λ^{∧} ∈ \overline{L}^{∧}.

Lift ψ(1 × σ) arbitrarily to ψ̃(1 × σ) and set, in general

ψ̃(z × σ) = ψ̃(z × 1)ψ̃(1 × σ).

Let

ψ̃(w_{1})ψ̃(w_{2}) = a(w_{1}, w_{2})ψ̃(w_{1}w_{2}),

where a(w_{1}, w_{2}) is a continuous 2-cocycle on W_{F} with values in Sô. What we have to do is show that there is a continuous function b(w) on W_{F} with values in Sô such that

b(s_{1})w_{1}(b(w_{2}))a(w_{1}, w_{2}) = b(w_{1}w_{2}).

What we do is introduce the extension K of topological groups defined by this cocycle and show that it splits continuously.

This is clear if F = C; so take F = R. Let N^{∧} be the lattice of rational characters of Sô. Consider the inverse image of C^{×} in K. This extension of C^{×} splits. Write an element in it as

s × z = e^{H} × e^{x}

with x in C and H in N ⊗ C. Let

σ(e^{H×z}) = e^{σ(H)+zμ+\overline{z}ν} × e\overline{z}

with μ and ν in N ⊗ C. Applying σ again we see that ν = −σ(μ). Moreover μ + σ(μ) must lie in N. In fact σ must fix the square of any lifting of 1 × σ to H. Since this square is of the form s × (−1),

e^{πi(μ+σ(μ))} = 1

and μ + σ(μ) ∈ 2N. Set

γ = \frac{μ − σ(μ)}{4}, δ = \frac{μ + σ(μ)}{4}

and

α = γ − δ, β = γ + δ.

Then μ = 2(γ + δ), α − β = −2δ lies in N and σ(α) + μ = β, σ(β) − σ(μ) = α. We replace the original splitting over C^{×} by

e^{z} → e^{zα+\overline{z}β} × e^{z}.

Since

σ(e^{zα+\overline{z}β} × e^{z}) = e^{z(σ(α)+μ)+\overline{z}(σ(β)−σ(μ))} × e\overline{z} = e^{zβ+\overline{z}α} × e\overline{z}

this new splitting is respected by the action of σ.

We have still to split the extension completely. Choose a representative h of 1×σ in H. Let h^{2} = s×(−1). Let S = e^{H} and H = H_{+} + H_{−}, with σ(H_{+}) = H_{+}, σ(H_{−}) = −H_{−}. Replacing h by

(e^{−H_{+}/2} × 1)h

if necessary, we may suppose that H_{+} = 0. Since σ(s) = s, 2H lies in 2πiN. Write

H = πi(λ − σ(λ))

with λ ∈ N ⊗ C. We may modify the splitting over C^{×} once again, replacing it by

e^{z} → e^{zλ+\overline{z}σ(λ)} × e^{z}.

In this new splitting over C^{×}, h^{2} is given by

e^{H−πiλ+πiσ(λ)} × −1 = 1 × −1.

We have now split the extension completely.

To show that χ_{φ} is independent of Q is easy. Suppose Q_{1}, Q_{2} together with ζ_{1}, ζ_{2} are two possible choices. Since we may replace the pair Q_{1}, Q_{2} by Q_{3}, Q_{1} or Q_{3}, Q_{2} with

Q_{3} = {(p_{1}, p_{2})|ζ_{1}(p_{1}) = ζ_{2}(p_{2})}

there is no harm in supposing that Q_{1} is given by a surjective homomorphism ξ : Q_{1} → Q_{2}. When this is so, Lemma 2.8 shows immediately that Q_{1} and Q_{2} give the same quasi-character χ_{φ}.

The following fact follows easily from the construction and Lemma 2.9.

LEMMA 2.11. The map φ → χ_{φ} respects restriction of scalars.

Let Z^{∧} be the center of Gô. The action of W_{F} on Z^{∧} is well defined and so is the group H^{1}(W_{F} , Z^{∧}), where it is understood that only continuous cocycles are to be considered. If φ ∈ Φ(G) and α ∈ H^{1}(W_{F} , Z^{∧}) define αφ by

αφ(w) = α(w)φ(w).

As is implicit in the notation and is easily verified the class of αφ depends only on that of α and φ. Thus the group H^{1}(W_{F} , Z^{∧}) acts on Φ(G). We should also be able to make it act in Π(G). To do this we associate to α a continuous homomorphism π_{α} of G(F) into C^{×}.

Let G_{der} be the derived group of G, G_{sc} the simply connected covering group of G_{der}, and G_{corad} the quotient of G by G_{der}. We have

G_{sc}
↓ ↘
1 → G_{der} → G → G_{corad} → 1.

Passing to associate groups we have

Gô ; sc

↑ ↖
1 ← Gô ; der ← Gô ← Gô ; corad ← 1

Suppose we have a diagram 1 ↓ B

↓ ↘
1 → G_{sc} → G̃ → D → 1
↘  ↓

G ↓ 1

in which the vertical and horizontal lines are exact, B and D are tori, and G̃(F) → G(F) is surjective. For example if R is the kernel of G_{sc} → G and

K = Hom(R ⊗_{F} \overline{F} , GL(1))

we could take a free 𝔤(\overline{F} /F) module P that maps surjectively to K, set L̃ equal to the group of pairs (λ, p), λ ∈ L_{sc}, p ∈ P, with the same image in K, and Δ̃ equal to {(α, 0)|α ∈ Δ}, and define G by means of L̃, Δ̃, and the twisting defining G. Passing to associate groups yields

1 ↑ Bô

↑ ↖
1 ← Gô ; sc ← G̃ô ← Dô ← 1
↖ ↑
Gô ← Z^{∧}

↑ This diagram gives Z^{∧} as the kernel of Dô → Bô and hence a map of H^{1}(W_{F} , Z^{∧}) into the kernel of H^{1}(W_{F} , Dô) → H^{1}(W_{F} , Bô). By Lemma 2.8 every element β of H^{1}(W_{F} , Dô) = Φ(D) yields a quasicharacter of D(F) and hence of G̃(F). It is trivial on B(F) and hence gives a quasi-character of G(F) if and only if β becomes zero in H^{1}(W_{F} , Bô). In particular every element α of H^{1}(W_{F} , Z^{∧}) yields a quasi-character π_{α} of G(F). If G̃_{1} and G̃_{2} are possible choices for G̃ so is G̃_{1} ×_{G} G̃_{2}. Using this, one shows easily that π_{α} does not depend on the choice of G̃.

LEMMA 2.12. The map α → π_{α} respects restriction of scalars. If φ′ = αφ then

χ′_{φ}(z) = π_{α}(z)χ_{φ}(z), z ∈ Z(F).

The first assertion follows easily from Lemma 2.9 and the construction. Suppose G_{1} is the group for which we are trying to prove the second assertion. Let G be quasi-split and let ψ be an isomorphism of G_{1} and G such that ψ^{−1}τ(ψ) is inner for τ ∈ 𝔤(\overline{F} /F). We may so construct G̃_{1} and G̃ that ψ lifts to ψ̃ : G̃_{1} → G̃. D_{1} and D will be the same and

\begin{aligned}G̃_{1} & → & D_{1} \\ ψ̃ ↓ & & ∥ \\ G̃ & → & D \end{aligned}

will be commutative. Since ψ restricted to Z_{G1} is defined over F and yields an isomorphism of Z_{G1} with Z_{G2} and since

χ_{φ}(ψ(z)) = χ_{φ}(z)

if φ ∈ Φ(G − 1) = Φ(G), we need only prove the lemma for G.

If T̃ and T are corresponding Cartan subgroups of G̃ and G, defined over F and lying in Borel subgroups over F, then

\begin{aligned}T̃ ô & ← & Dô \\ ↑ & & ↑ \\ T ô & ← & Z^{∧} \end{aligned}

is commutative. Thus on T (F), π_{α} is the quasi-character defined by the image of α in

H^{1}(W_{F} , T ô) = Φ(T).

Although we do not need to know it, it could be observed that T̃(F) → D(F) is surjective because H^{1}(𝔤(\overline{F} /F), T_{sc}) = 0. Thus π_{α} is determined by its values on T (F).

Now consider the objects used to define χ_{φ}. We had a surjection Φ(\overline{G}) → Φ(G). If φ is the image of \overline{φ} then χ_{φ} is determined by the image of \overline{φ} in Φ(\overline{G}^{∧}_{rad}). But \overline{G}ô ; rad = \overline{T} ô ; rad and, by Lemma 2.10, Φ(\overline{T}) → Φ(\overline{T}_{rad}) is surjective. Thus for φ ∈ Φ(G) there is an η that lies in the image of Φ(T) → Φ(G) and lifts to \overline{η} in the image of Φ(\overline{T}) → Φ(\overline{G}) such that \overline{η} and \overline{φ} have the same image in Φ(\overline{G}_{rad}). Then χ_{φ} = χ_{η}. If η is the image of β in Φ(T) then, by construction almost, χ_{η} is the restriction to Z_{G}(F) of the quasi-character of T (F) associated to β. Since \overline{Z}^{∧}, the center of \overline{G}ô, is the inverse image of Z^{∧} in \overline{G}ô we may choose η′ corresponding to φ′ to be the image of γβ, if γ is the image of α ∈ H^{1}(W_{F} , Z^{∧}), in H^{1}(W_{F} , T ô) = Φ(T). The lemma now follows.

Notice that if ψ : H → G is defined over F and has an abelian kernel and an abelian cokernel then we can associate to it a homomorphism ψ^{∧} : G^{∧} → H^{∧}.

3. The definitions. The group G(F) is a Lie group. Let 𝔤 be the tensor product of its Lie algebra with C, let 𝔄 be the universal enveloping algebra of 𝔤 and let ℨ be its center. A (continuous) representation π of G(F) on a Banach space V will be called irreducible if V contains no nontrivial closed invariant subspaces; it will be called quasi-simple if the elements of ℨ act on the infinitely differentiable vectors as scalars.

Let π be irreducible and quasi-simple. Let K be a maximal compact subgroup of G(F) and let μ and ν be irreducible representations of K on the finite-dimensional spaces X and Y. Suppose we have K-homomorphisms ζ and η of X and Y into V and its dual V^{∗} respectively. Suppose moreover that ζ(x) is infinitely differentiable for all x ∈ X. Let Ψ = Ψ_{ζ,η} be the function on G(F) with values in X^{∗} ⊗ Y^{∗} defined by

Ψ(g) : (x, y) → ⟨π(g)ζ(x), η(y)⟩.

Then Ψ is a spherical function on G(F) of type μ^{∗}, ν^{∗}, if μ^{∗} and ν^{∗} are contragredient to μ, ν. If we regard the elements of 𝔄 as left-invariant differential operators on G(F) then

ZΦ = κ(Z)Ψ, Z ∈ ℨ,

if π(Z) = κ(Z)I. Because π is quasi-simple and irreducible, π(z) is a scalar for z ∈ Z_{G}(F) and

Ψ(gz) = π(z)Ψ(g).

If G^{o}(F) is the connected component of G(F) then

G(F) = KG^{o}(F) = KZ_{G}(F)G_{der}^{o}(F)

(cf. [11]); so Ψ is determined by its restriction to G_{der}^{o}(F). Notice also that if v ∈ V, v^{∗} ∈ V^{∗}, and ⟨π(g)v, v^{∗}⟩ = 0 for all g then either v or v^{∗} is zero.

It follows from these considerations and Proposition 9.1.3.1 of [16] that

dim Hom_{K}(X, V) < ∞,

so that any representation of K occurs with finite multiplicity in V. Let V_{K} be the space of K-finite vectors. Every vector in V_{K} is infinitely differentiable so that both 𝔄 and K operate on V_{K}. The representations π and π′ on V and V′ are said to be infinitesimally equivalent if the representations of the pair 𝔄, K on V′_{K} are algebraically equivalent. Since any two maximal compact subgroups of G(F) are conjugate, this notion does not depend on the choice of K. Π(G) will be the set of infinitesimal equivalence classes of irreducible quasi-simple representations of G(F). We shall usually not distinguish between a representation and its class.

To every φ in Φ(G) we are going to associate a finite but nonempty set Π_{φ} in Π(G) such that the following conditions are valid.

(i) If φ ≠ φ′ then Π_{φ} and Π′_{φ} are disjoint.

(ii) If π ∈ Π_{φ} then

π(z) = χ_{φ}(z)I, z ∈ Z_{G}(F).

(iii) If φ′ = αφ with α ∈ H^{1}(W_{F} , Z^{∧}) then

Π′_{φ} = {π_{α} ⊗ π|π ∈ Π_{φ}}.

(iv) If η : H → G has abelian kernel and cokernel, if φ ∈ Φ(G) and φ′ = η′(φ), then the pullback of any π ∈ Π_{φ} to H(F) is the direct sum of finitely many irreducible, quasi-simple representations, all of which lie in Π_{φ′}.

(v) If φ ∈ Φ(G) and one element of Π_{φ} is square integrable modulo Z_{G}(F) then all elements are. This happens if and only if φ(W_{F}) is contained in no proper parabolic subgroup of G^{∧}.

We remark that the representation π is said to be square integrable modulo the center if π = ζ ⊗ π′ where ζ is one-dimensional and where π′, which operates on V′, is such that | f′(π′(g)v′) |^{2} is an integrable function on Z_{G}(F)\G(F) for any K-finite v′ ∈ V′ and any K-finite linear form f′ on V′.

(vi) If φ ∈ Φ(G) is tempered then all elements are. With respect to a distinguished splitting, write φ(w) = a(w) × w. The elements of Π_{φ} are tempered if and only if {a(w)|w ∈ W_{F} } is relatively compact in G^{∧}.

π, acting on V, is said to be tempered if f(π(g)v) satisfies the weak inequality for any K-finite v ∈ V and any K-finite linear form f on V.

Since we can always restrict scalars, we may as well take F to be R. Let φ ∈ Φ(G). Let A be the Zariski-closure of the image of W_{R} under the composition of φ with the homomorphism of G^{∧} into the group of automorphism of 𝔤^{∧}, the Lie algebra of Gô. Let B be the Zariski-closure of the image of C^{×}. Since the elements in the image of C^{×} commute and are, by assumption, semisimple they can be simultaneously diagonalized. Thus every element of B is semisimple. The same is true for A, because A^{2} ⊆ B. Since A is clearly supersolvable we may apply Theorem 5.16 of [13] to see that φ(W_{R}) normalizes a Cartan subgroup Sô in Gô. Since the group of automorphisms of Sô is discrete, φ(C^{×}) must centralize Sô. Consequently 𝔤(C/R) = C^{×}\W_{R} acts on Sô, on M^{∧} = Hom(Sô, C^{×}), on M = Hom(M^{∧}, Z), and on M ⊗ R.

Suppose 𝔤(C/R) fixes a point λ in M ⊗R. If P^{∧} is the parabolic subgroup of G^{∧} defined by the condition that α^{∧} is a root of Sô in P ô if and only if ⟨λ, α^{∧}⟩ ≥ 0 then φ(W_{R}) lies in P^{∧}.

We shall first define Π_{φ} under the assumption that φ(W_{R}) is contained in no proper parabolic subgroup of G^{∧}. Then if λ is fixed by 𝔤(C/R)

⟨λ, α^{∧}⟩ = 0

for all α^{∧}.

LEMMA 3.1. If Φ(G) contains a φ with the property that φ(WR) is contained in no proper parabolic subgroup of G^{∧} then G_{der} has a Cartan subgroup T_{der} with T_{der}(F) compact.

We have a map G^{∧} → G^{∧}_{der} that yields Φ(G) → Φ(G_{der}). Replacing φ by its image in Φ(G_{der}), we may suppose that G = G_{der}. Then M^{∧} ⊗ R is spanned by the roots of α^{∧}. Consequently the nontrivial element σ in 𝔤(C/R) fixes no element of M ⊗ R but 0 and act as −1.

Let ψ be an isomorphism of G with a quasi-split group G′. Choose a Borel subgroup B′ of G′ over R and a Cartan subgroup T′ of B′. Use G′, B′, and T′ to build G^{∧}, Bô and T ô as a concrete realization of the associate group. Replacing φ by another homomorphism in the same class, we may suppose that Sô = T ô, so that M^{∧} = L^{∧}. There are, however, two actions of σ on L^{∧}, the one built into the construction of G^{∧}, which we denote by λ^{∧} → σλ^{∧}, and the one defined by φ, which we denote by λ^{∧} → \overline{σ}λ^{∧}. There is an ω^{∧} in Ω^{∧} such that

\overline{σ}λ^{∧} = ω^{∧}σλ^{∧}.

Since \overline{σ} acts as −1, σ commutes with \overline{σ} and ω^{∧}. Let ω be the element of Ω′, the Weyl group of T′, corresponding to ω^{∧}.

Because any two quasi-split groups differing by an inner twisting are isomorphic, we may suppose that σ acts on G′(C) in such a way that

σλ^{∧}(στ) = σ(λ^{∧}(t)), t ∈ T′(C),

and

σ(X′_{α}) = X′_{σα}, α ∈ Δ′.

Here the X′_{α} are appropriately chosen root vectors in the Lie algebra of G. Define X_{−α} so that if H_{α} = [X_{α}, X_{−α}] then α(H_{α}) = 2. The algebraic automorphism of G′ defined by t → ω(t), X_{α} → X_{ω(α)} commutes with the action of σ and its square is 1. It is of course inner. We use the cocycle a_{1} = 1, α_{σ} = ω to twist G′ and obtain G′′. G′′ contains the Cartan subgroup T′′ obtained by twisting T′. Since

λ(ωσt) = σ(λ(t))^{−1}

T′′(R) is compact.

There is an isomorphism η of G′′ with G such that ξ = η^{−1}σ(η) is inner. We may suppose that η(T′′) = T is a Cartan subgroup of G over R for which the compact part of T (R) has maximal dimension. What we have to do to show that T (R) is compact is to show that ξ, which normalizes T′′, actually centralizes T′′, for then η : T′′ → T is defined over R. We use an idea that can be found in many places. If t ∈ T′′(C) and λ is a rational character of T′′ then λ(σ(t)) = σ(λ(t)^{−1}). Thus

λ(σξ(t)) = σ(λ(ξt)^{−1}) = σ(ξ^{−1}λ(t)^{−1}) = ξ^{−1}λ(σ(t)) = λ(ξσ(t))

and ξσ = σξ. Since ξσ = 1, ξ^{2} = 1.

Suppose α is a root of T′′ and ξα = −α. Consider the subgroup H′′ of G′′ that is generated by T′′(C) and the one-parameter subgroups exp zX′′_{α}, exp zX′′_{−α}, z ∈ C. H′′ is invariant under σ and ξ and H = η(H′′) is defined over R. I claim that H_{sc} is isomorphic to SL(2) over R and that T_{sc}, the inverse image of T in H_{sc} is the Cartan subgroup whose set of real points is noncompact. This clearly contradicts the definition of T and shows that ξα ≠ −α for all α. To prove the assertion about H_{sc}, we start from the observation that we may choose X′′_{α} and X′′_{−α} so that [X′′_{α}, X′′_{−α}] = −H′′_{α} with α(H′′_{α}) = 2 and so that σ(X′′_{α}) = X′′_{−α}. Then σ(X′′_{−α}) = X′′_{α}. Let ξ(X′′_{α}) = aX′′_{−α}, ξ(X′′_{−α}) = bX′′_{α}. Then

[aX′′_{−α}, bX′′_{α}] = −H′′_{α};

so ab = 1. However the relation ξσ(ξ) = 1 shows that a\overline{a} = b\overline{b} = 1. Recall that, on C, σ is complex conjugation. Choose s in T′′(C) such that

α(s)^{−1}\overline{α(s)} = a.

Replacing η by η ∘ ads, we suppose that a = b = 1. Set

H_{α} = η(H′′_{α}), X_{α} = η(X′′_{α}), X_{−α} = −η(X′′_{α}).

Then

σ(X_{α}) = η(ξσ(X′′_{α})) = η(X′′_{α}) = X_{α},
σ(X_{−α}) = X_{−α},
σ(H_{α}) = H_{α}.

Thus H_{α}, X_{α}, X_{−α} span a Lie algebra that, together with the action of σ on it, is isomorphic to the Lie algebra of SL(2). Since H_{α} lies in the Lie algebra of T, this gives the required assertion.

Let M′′ be the lattice of rational characters of T′′. Since α + ξα is different from 0 for all α there is a point H in the dual of M′′ ⊗ R such that

⟨α + ξα, H⟩ = ⟨α, H + ξH⟩ ≠ 0

for all α. But ξ fixes H + ξH and therefore fixes the chamber in which it lies. Since ξ is inner this is possible only if ξ centralizes T′′.

The lemma proved, we return to the original G and φ. Although it is not important, we choose for the sake of definiteness an isomorphism ψ of G with a quasi-split group G′, with ψ^{−1}σ(ψ) inner, choose B′ and T′, construct Gô, Bô, T ô accordingly, and take the associate group to be G^{∧} = Gô × W_{R}. We also suppose that φ(W_{R}) normalizes T ô. Write φ(w) = a(w) × w. If z ∈ C^{×} then a(z) ∈ T ô. Let

λ^{∧}(a(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}.

If λ^{∧} → \overline{σ}λ^{∧} denotes the action of σ on L^{∧} defined by φ then ν = \overline{σ}μ, μ = \overline{σ}ν. Also if a = a(1 × σ),

λ^{∧}(aσ(a)) = λ^{∧}(a(−1)) = (−1)^{⟨μ−ν,λ^{∧}⟩}.

If ⟨α, λ^{∧}⟩ = 0 for all roots α then λ^{∧} is a rational character of Gô and we may define λ^{∧}(a). Notice in particular that ⟨α, λ^{∧}⟩ = 0 for all roots α if λ^{∧} = \overline{σ}λ^{∧}. The next lemma is critical.

LEMMA 3.2. Suppose h = a × w, with w = 1 × σ, lies in G^{∧}, normalizes T ô, and hα^{∧} = −α^{∧} for every root α^{∧}. Then aσ(a) ∈ T ô and, if δ is one-half the sum of the positive roots with respect to any order,

λ^{∧}(aσ(a)) = (−1)^{⟨2δ,λ^{∧}⟩}μ^{∧}(a) = (−1)^{⟨δ−hδ,λ^{∧}⟩}μ^{∧}(a)

if μ^{∧} = λ^{∧} + hλ^{∧}.

Of course an h satisfying the conditions of the lemma does not always exist. When it does a is any element of the normalizer of T ô in Gô that takes positive roots to negative roots.

That aσ(a) ∈ T ô and that 2δ = δ − hδ is clear. If s ∈ T ô and h is replaced by sh then aσ(a) is replaced by

sh(s)aσ(a)

where h(s) = hsh^{−1}. Since μ^{∧}(a) becomes

μ^{∧}(sa) = λ^{∧}(s)hλ^{∧}(s)μ^{∧}(a)

and

λ^{∧}(sh(s)) = λ^{∧}(s)hλ^{∧}(s)

we are free to replace h by sh. Thus we may suppose that a ∈ Gô ; ss or, more simply, that G, and hence Gô, is semisimple. Since it only makes the matter more difficult we may then replace G By G_{ad} and Gô by Gô ; ad, which is simply connected. Then the whole situation factors and we may finally assume that Gô is simple and simply connected.

Suppose β^{∧} is the largest root with respect to the given order [2]. Then σ(X_{β^{∧}}) = ηX_{β^{∧}} with η = ±1. If σ acts trivially on Gô then η = 1. In general I claim that if

β^{∧} = ∑_{α^{∧}∈Δ^{∧}} n(α^{∧})α^{∧}

is the expression of β^{∧} as a sum of simple roots and if l is one-half the sum of those n(α^{∧}) for which α^{∧} ≠ σα^{∧} and (α^{∧}, σα^{∧}) ≠ 0 then η = (−1)^{l}. This statement is not true for β^{∧} alone but for any positive root

γ^{∧} = ∑ m(α^{∧})α^{∧}

fixed by σ. Of course n(α^{∧}) is to be replaced by m(α^{∧}) and η by η(γ^{∧}), where

σ(X_{γ^{∧}}) = η(γ^{∧})X_{γ^{∧}}.

We prove it by induction on m = ∑ m(α^{∧}).

If m = 1 then l = 0; but by construction η(γ^{∧}) = 1. Suppose m > 1, so that γ^{∧} is not simple. Choose a simple root α1^{∧} such that (γ^{∧}, α1^{∧}) > 0. If α2^{∧} = σα1^{∧}, then (γ^{∧}, α2^{∧}) = (γ^{∧}, α1^{∧}). If α1^{∧} = α2^{∧} then \overline{γ}^{∧} = γ^{∧} − α1^{∧} is also a root and

X_{γ^{∧}} = [X_{α_{1}^{∧}} , X\overline{γ}^{∧}];

so η(γ^{∧}) = η(\overline{γ}^{∧}). Moreover l(γ^{∧}) = l(\overline{γ}^{∧}). If α1^{∧} ≠ α2^{∧} and (α1^{∧}, α2^{∧}) = 0 then \overline{γ}^{∧} = γ^{∧} − α1^{∧} − α2^{∧} is a root. The integers l(γ^{∧}) and l(\overline{γ}^{∧}) are equal. Since

X_{γ^{∧}} = [X_{α_{1}^{∧}} , [X_{α_{2}^{∧}} , X\overline{γ}^{∧}]] = [X_{α_{2}^{∧}} , [X_{α_{1}^{∧}} , X\overline{γ}^{∧}]],

η(γ^{∧}) = η(\overline{γ}^{∧}). If (α1^{∧}, α2^{∧}) ≠ 0 then α^{∧} = α1^{∧} + α2^{∧} is a root and

X_{α^{∧}} = [X_{α_{1}^{∧}} , X_{α_{2}^{∧}}];

so η(α^{∧}) = −1. If γ^{∧} = α^{∧} we are done. Otherwise \overline{γ}^{∧} = γ^{∧} − α^{∧} is a root, l(γ^{∧}) = l(\overline{γ}^{∧}) + 1, and η(γ^{∧}) = −η(\overline{γ}^{∧}) because

X_{γ^{∧}} = [X_{α^{∧}}, X\overline{γ}^{∧}].

Since (β^{∧}, α^{∧}) ≥ 0 for all positive roots, every root perpendicular to β^{∧} is a linear combination of simple roots perpendicular to it. Let Hô be the connected subgroup of Gô corresponding to the Lie algebra generated by {X_{α^{∧}}|(α^{∧}, β^{∧}) = 0}. H^{∧} = Hô × W_{R} ⊆ G^{∧} is also an associate group and we may assume the lemma has been proved for it. Let J ô be the group corresponding to the Lie algebra generated by X_{β^{∧}}, X_{−β^{∧}}. J ô is also invariant under W_{R}. The groups Hô and J ô commute with each other. Let a_{1} be an element of Hô normalizing T ô and taking positive roots in Hô to negative roots. Let a_{2} be an element of J ô normalizing T ô and taking β^{∧} to −β^{∧}. a_{1} fixes β^{∧}. Thus if α^{∧} is positive and (α^{∧}, β^{∧}) > 0, then (a_{1}α^{∧}, β^{∧}) > 0 and a_{1}α^{∧} is positive. But (a_{2}α^{∧}, β^{∧}) = −(α^{∧}, β^{∧}) < 0 so a_{2}α^{∧} is negative. The product a_{1}a_{2} takes every positive root to a negative root and we may take a = a_{1}a_{2}. Since a_{2} centralizes Hô, a_{1} × (1 × σ) in H^{∧} takes every positive root to its negative.

By induction

λ^{∧}(a_{1}σ(a_{1})) = (−1)∑Γ_{0}^{(α,λ^{∧})}

if Γ_{0} = {α > 0 | ⟨α, β^{∧}⟩ = 0}. J ô is covered by SL(2, C). We may suppose that

( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ) → X_{β^{∧}}, ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ) → X_{−β^{∧}.}

Then the action of σ lifts to conjugation by

( \begin{matrix}1 & 0 \\ 0 & (−1)^{l} \end{matrix} ) .

Since we may take a_{2} to be the image of

( \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} )

a_{2}σ(a_{2}) is the image of

( \begin{matrix}(−1)^{l+1} & 0 \\ 0 & (−1)^{l+1} \end{matrix} ) .

Thus

λ^{∧}(a_{2}σ(a_{2})) = (−1)^{(l+1)⟨β,λ^{∧}⟩}.

To prove the lemma we have to show that

l⟨β, λ^{∧}⟩ ≡ ∑_{{a>0 \atop ⟨α,β^{∧}⟩≠0} ; α≠β} ⟨α, λ^{∧}⟩ (mod 2).

If α > 0, ⟨α, β^{∧}⟩ ≠ 0, and if α ≠ β then β⟨α, β^{∧}⟩ − α is also a positive root and is different from α. Thus the right side is l′⟨β, λ^{∧}⟩ if

l′ = \frac{1}{2} ∑_{{a>0 \atop a≠β} ; ⟨α,β^{∧}⟩≠0}⟨α, β^{∧}⟩ = \frac{1}{2}\{∑_{α>0}⟨α, β^{∧}⟩\} − 1 = ⟨δ, β^{∧}⟩ − 1 = ∑ n(α^{∧}) − 1.

It would be enough to show that l = l′(mod2).

To finish up we make use of some standard facts [2]. The order h of a Coxeter element is l′ + 1. If σ acts trivially then l = 0. But if σ acts trivially then a itself must take every root to its negative. This forces l′ + 2 to be even ([2], p. 173). If σ does not act trivially the roots are all of the same length. There is an α in Δ such that ⟨α, β^{∧}⟩ = 1 ([2], p. 165). Since ⟨2δ, α^{∧}⟩ = 2

α^{∧}(aσ(a)) = (−1)^{l−l′}.

However h acts on the Lie algebra of Gô, and

h(X_{α^{∧}}) = cX_{−α^{∧}}, h(X_{−α^{∧}}) = dX_{α^{∧}}, h([X_{α^{∧}}, X_{−α^{∧}}]) = −[X_{α^{∧}}, X_{−α^{∧}}].

This forces cd to be 1; so

α^{∧}(aσ(a))X_{α^{∧}} = h^{2}(X_{α^{∧}}) = X_{α^{∧}}

and l − l′ is even.

There is another lemma to be verified before we can define Π_{φ}.

LEMMA 3.3. Suppose φ ∈ Φ(G), φ(W_{R}) is contained in no proper parabolic subgroup of G^{∧}, φ(W_{R}) normalizes T ô, and

λ^{∧}(φ(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}, z ∈ C^{×}.

Then ⟨μ, α^{∧}⟩ is different from 0 for all roots α^{∧}. Moreover φ(W_{R}) normalizes no other Cartan subgroup of Gô.

Suppose ⟨μ, α^{∧}⟩ = 0. X_{α^{∧}} is fixed by φ(z), z ∈ C^{×}. Let h = φ(1 × σ) and set
U = X_{α^{∧}} + h(X_{α^{∧}}).

Then hU = U. Let 𝔥 be the space of vectors H in the Lie algebra of T ô for which α^{∧}(H) = 0. Then 𝔥 + CU is the Lie algebra of a Cartan subgroup of Gô normalized by φ(W_{R}). It is however, clear that the action of σ on the roots does not take every root to its negative. This we know is incompatible with the assumption that φ(W_{R}) is contained in no proper parabolic subgroup of G^{∧}. Thus ⟨μ, α^{∧}⟩ is never 0 and the centralizer of φ(C^{×}) in the Lie algebra of Gô is exactly the Lie algebra of T ô. The second assertion of the lemma follows.

If n ∈ Gô and

φ′ : w → nφ(w)n^{−1}

satisfies the conditions of the lemma, with μ, ν replaced by μ′, ν′, then n must normalize T ô and μ′ = nμ, ν′ = nν. Consequently the orbit of μ under the Weyl group is determined by the class of φ alone. Since μ − ν ∈ L and

⟨μ − ν, α^{∧}⟩ = ⟨μ − \overline{σ}μ, α^{∧}⟩ = 2⟨μ, α^{∧}⟩

the number ⟨μ, α^{∧}⟩ is real for all α^{∧}. Since it is different from 0, we may choose φ in its class so that ⟨μ, α^{∧}⟩ > 0 for all α^{∧} ∈ Δ^{∧}. This done, the only way we may modify φ is to replace it by

φ′ : w → sφ(w)s^{−1}

with s ∈ T ô.

We have observed that if ⟨α^{∧}⟩ = 0 for all roots α, we may define λ^{∧}(a), where φ(1 × σ) = a × (1 × σ). Choose λ_{0} ∈ L ⊗ C such that

λ^{∧}(a) = e^{2πi⟨λ_{0},λ^{∧}⟩}

for such λ^{∧}. λ_{0} is determined modulo

L + ∑_{α∈Δ} C_{α} + \{λ − \overline{σ}λ | λ ∈ L ⊗ C\}

or, since α = (α − \overline{σ}α)/2,

L + \{λ − \overline{σ}λ | λ ∈ L ⊗ C\}.

We know that ⟨μ, α^{∧}⟩ is real and different from 0 for all α. Let

δ = \frac{1}{2} ∑_{⟨μ,α^{∧}⟩>0} α

and set μ_{1} = μ − δ, ν_{1} = \overline{σ}μ_{1} = ν + δ. Then μ_{1} − ν_{1} = μ − ν − 2δ ∈ L, because μ − ν ∈ L. If λ^{∧} ∈ L^{∧} then, by Lemma 3.2,

e^{πi⟨μ−ν,λ^{∧}⟩} = λ^{∧}(aσ(a)) = e^{πi⟨2δ,λ^{∧}⟩+⟨λ_{0}+\overline{σ}λ_{0},λ^{∧}⟩};

so

λ_{0} + \overline{σ}λ_{0} ≡ \frac{μ_{1} − ν}{2} (modL).

Thus

⟨μ, α^{∧}⟩ = \frac{1}{2}⟨μ − \overline{σ}μ, α^{∧}⟩ = \frac{1}{2}⟨μ_{1} − ν_{1}, α^{∧}⟩ + ⟨δ, α^{∧}⟩
≡ ⟨λ_{0} + \overline{σ}λ_{0}, α^{∧}⟩ + ⟨δ, α^{∧}⟩ = 1

modulo Z and ⟨μ, α^{∧}⟩ is integral. If ⟨μ, α^{∧}⟩ > 0 then

⟨μ_{1}, α^{∧}⟩ = ⟨μ, α^{∧}⟩ − ⟨δ, α^{∧}⟩ = ⟨μ, α^{∧}⟩ − 1 ≥ 0.

Let S be a Cartan subgroup of G over F for which S(R) ∩ G_{der}(R) is compact. We may choose the isomorphism ψ of G with G′ so that ψ(S) = T′. The isomorphism allows us to identify L, the lattice of rational characters of T′, with the lattice of rational characters of S. Then the semidirect product T ô×𝔤(C/R) with σ acting on T ô as \overline{σ} becomes the associate group of S. As in the proof of Lemma 2.8, μ_{1}, ν_{1}, and λ_{0} then define a homomorphism χ of S(R) into C^{×}. However since ψ is not uniquely determined, μ_{1}, ν_{1} and λ_{0} are only determined modulo the action of the Weyl group. Although the elements of the Weyl group of S may be represented by elements of G(C) their action on S is defined over R. If we replace μ_{1}, ν_{1} and λ_{0} by ωμ_{1}, ων_{1} = \overline{σ}ωμ_{1}, and ωλ_{0}, which is congruent to λ_{0} modulo \{λ − \overline{σ}λ | λ ∈ L ⊗ C\}, then χ is replaced by χ′ : t → χ(ω^{−1}(t)). Let X_{φ} be the set of these quasi-characters χ. X_{φ} is determined by the class of φ alone.

To verify that the sets Π_{φ}, which we shall soon define, are disjoint and exhaust the classes of representations of G(R) that are square-integrable modulo the center, we shall need the following lemma.

LEMMA 3.4. Suppose G has a Cartan subgroup S over R such that S(R) ∩ G_{der}(R) is compact and X is an orbit of the Weyl group in the set of quasi-characters of S(R). Then there is a unique φ ∈ Φ(G) such that φ(W_{R}) is contained in no proper parabolic subgroup of G^{∧} and such that X = X_{φ}.

We have first to observe that an h in G^{∧} that satisfies the conditions of Lemma 3.2 exists. Let ψ be an isomorphism of G with a quasi-split group G′ such that ψ^{−1}σ(ψ) is inner. If B′ and T′ are chosen as usual, we may suppose that ψ(S) = T′. We have to show that there is an element ω^{∧} of the Weyl group of T ô such that ω^{∧}σα^{∧} = −α^{∧} for all α^{∧}. This is equivalent to showing that there is an element ω of the Weyl group of T′ such that ωσα = −α for all α. ω and ω^{∧} will then be corresponding elements. Let ψσ(ψ^{−1}) = adn, with n in the normalizer of T′, then

−ψ^{−1}(α) = σ(ψ^{−1}(α)) = ψ^{−1}(ψσ(ψ^{−1})σα) = ψ^{−1}(nσα)

and nσα = −α. We take ω to be the element of the Weyl group represented by n.

Let \overline{σ} be the action on T ô, L^{∧}, and L determined by such an h. We regard T ô, with the action \overline{σ}, as the connected component of the associate group of S. If χ ∈ X choose μ_{1}, ν_{1} and λ_{0} so that

χ(t) = e^{⟨λ_{0},H−\overline{σ}\overline{H}⟩+⟨μ_{1}/2,H+\overline{σ}\overline{H}⟩}

if t = e^{H} lies in S(R). Given X, μ_{1}, ν_{1}, and λ_{0} are determined modulo the action of the Weyl group. Also μ_{1} − ν_{1} ∈ L and, since \overline{σ}α^{∧} = −α^{∧},

⟨μ_{1} − ν_{1}, α^{∧}⟩ = ⟨μ_{1} − \overline{σ}μ_{1}, α^{∧}⟩ = 2⟨μ_{1}, α^{∧}⟩

is real. Choose an order on the roots so that ⟨μ_{1}, α^{∧}⟩ ≥ 0 if α^{∧} is positive; let δ be one-half the sum of the positive roots α with respect to this order, and set μ = μ_{1} + δ, ν = \overline{σ}μ. Since the δ which arise in this way differ by an element of the Weyl group that fixes μ_{1}, the orbit of μ, ν, and λ_{0} under the Weyl group is determined by X_{φ} alone. The various μ are certainly nonsingular. To be definite choose the unique one that is positive with respect to Bô.

If φ is normalized in the way described earlier, it is clear that X = X_{φ} only if

λ^{∧}(φ(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}, z ∈ C^{×}.

Fix an h = b × (1 × σ) satisfying the conditions of Lemma 3.2 and choose a representative λ_{0}. If X = X_{φ} and φ gives rise to this particular λ_{0} then φ(1 × σ) = a × (1 × σ) with a = sb, s ∈ T ô, and

e^{2πi⟨λ_{0},λ^{∧}⟩} = λ^{∧}(a) = λ^{∧}(s)λ^{∧}(b)

if ⟨α, λ^{∧}⟩ = 0 for all α. An s in T ô satisfying this condition always exists. We will be able to extend φ to W_{R} if

λ^{∧}(aσ(a)) = e^{πi⟨μ−ν,λ^{∧}⟩}

for all λ^{∧}. By Lemma 3.2 the left side is

e^{2πi⟨δ,λ^{∧}⟩+2πi⟨λ_{0}+\overline{σ}λ_{0},λ^{∧}⟩}.

Since

\frac{μ_{1} − ν_{1}}{2} ≡ λ_{0} + \overline{σ}λ_{0} (modL)

it equals the right side. As for the uniqueness, if λ^{∧}(s) = 1 whenever ⟨α, λ^{∧}⟩ = 0 then, in particular, λ^{∧}(s) = 1 when \overline{σ}λ^{∧} = λ^{∧} and s = t\overline{σ}(t^{−1}) with t ∈ T ô. Then sh = tht^{−1}. Choosing a different representative for λ_{0} forces the same kind of change in s; so the class of φ is determined uniquely by X.

Suppose φ(W_{R}) is contained in a parabolic subgroup P^{∧} of G^{∧}. Then φ(W_{R}) is contained in a Levi factor M^{∧} of P^{∧} and normalizes a Cartan subgroup of M ô. But μ is regular; so T ô is the only Cartan subgroup centralized by φ(C^{×}) and therefore the only Cartan subgroup normalized by φ(W_{R}). P^{∧} must then contain T ô. Since φ(1 × σ) takes each root to its negative, P^{∧} is G^{∧}.

Suppose S and S′ are two Cartan subgroups of G such that S(R) ∩ G_{der}(R) and S′(R) ∩ G_{der}(R) are compact. There is a 𝔤 ∈ G(C) such that adg(S) = S′. The restriction of adg to S is defined over R and

X_{φ} = {χ ∘ adg′ | χ ∈ X′_{φ}}.

If g ∈ G(R) then g = g_{1}g_{2} where g_{1} lies in G_{der}^{0}(R) and g_{2} lies in the normalizer of S. If Ω is the Weyl group of S and Ω_{1} consists of those elements in Ω with a representative in G_{der}^{o}(R), the connected component of G_{der}(R), and ω is the image of g_{2} in Ω then g → Ω_{1}ω is a well-defined map of G(R) into Ω_{1}/Ω. The inverse image of Ω_{1} in G(R) is

G_{0}(R) = S(R)G_{der}^{0}(R) = S(R)G^{0}(R) = Z_{G}(R)G_{der}^{0}(R).

If χ ∈ X_{φ} and μ_{1} is defined above, choose an order on the roots so that ⟨μ_{1}, α^{∧}⟩ ≥ 0 if α is positive and let δ be one-half the sum of the roots positive with respect to this order. By Harish-Chandra’s theory of the discrete series there exists for each such pair χ, δ a unique irreducible representation π_{0}(χ, δ) of G_{0}(R), square-integrable modulo the center, whose character on the set of regular elements in S(R) is

ε(G) ∑_{ω∈Ω_{1}} \frac{sgnωχ(ωs)e^{δ(ωH−H)}}{Δ(s)} .

Here s = e^{H} is a regular element in S(R) and

Δ(s) = ∏_{⟨δ,α^{∧}⟩>0}(1 − e^{−α(H)}).

ε(G), which is ±1, depends on G alone.

It is clear that, if χ ∈ X_{φ}, χ′ ∈ X′_{φ}, the representations π_{0}(χ, δ) and π_{0}(χ′, δ′) are equivalent if and only if φ = φ′ and there is an ω ∈ Ω_{1} such that δ = ωδ′ and

χ′(s) = χ(ωs)

for all s ∈ S(R). If g ∈ G(R) has image Ω_{1}ω in Ω_{1}\Ω, π_{0} = π_{0}(χ, δ), and

π′_{0}(h) = π_{0}(ghg^{−1}), h ∈ G_{0}(R),

then π′_{0} = π_{0}(χ′, δ′) with ωδ′ = δ and χ′(s) = χ(ωs). Thus the representations

π(χ, δ) = Ind(G(R), G_{0}(R), π_{0}(χ, δ))

are irreducible. We set

Π_{φ} = {π(χ, δ) | χ ∈ X_{φ}}.

If the image of G(R) in Ω_{1}\Ω has e elements, then Π_{φ} contains

[Ω : Ω_{1}]/e

classes.

Before explaining why conditions (i) to (vi) are, insofar as they apply to the Π_{φ} already defined, fulfilled, we verify a simple lemma.

LEMMA 3.5. The restriction of an irreducible quasi-simple representation π of G(R) to G_{der}^{σ}(R) is infinitesimally equivalent to the direct sum of finitely many irreducible representations of G_{der}^{o}(R).

Let π act on V. Let K be a maximal compact subgroup of G(R) and let K^{o} be K ∩ G_{der}^{o}(R). Since π(z) is a scalar for z ∈ Z_{G}(R) and since

[K : K_{0}(K ∩ Z_{G}(R))] < ∞

every irreducible representation of K_{0} occurs with finite multiplicity in the restriction of π to K_{0}. Let ρ be an irreducible representation which actually occurs and g_{1} …, g_{r} be a complete set of representatives for the cosets of Z_{G}(R)G_{der}^{o}(R) in G(R). If ξ is the character of ρ let U_{j} be the range of the projection

E_{j} = ∫_{K_{0}} ξ(k)π(g_{j}^{−1}kg_{j})dk.

U_{j} is finite-dimensional. Let E be

∫_{K_{0}} ξ(k)π(k)dk.

If U is a closed nonzero subspace of V invariant under G_{der}^{o}(R) then U ∩ (∑_{j} U_{j}) is not zero. If it were

0 = π(g_{j})E_{j}U = Eπ(g_{j})U

for each j. But V is contained in the closure of

∑ π(g_{j})U

and EV ≠ 0. As a consequence, among the closed nonzero subspaces of V invariant under G_{der}^{o}(R) there is at least one minimal one W. The representation of G_{der}^{o}(R) on W is irreducible.

Choose a maximal collection h_{1} …, h_{l} from {g_{1} …, g_{r}} such that

∑_{i=1}^{l} π(h_{i})W

is direct. Each π(h_{1})W is invariant and irreducible under G_{der}^{o}(R). Moreover

⨁_{i=1}^{l} π(h_{i})W

is dense in V and therefore contains all K-finite vectors. The lemma follows.

If Ω_{2}(G) is the group of elements in the Weyl group of S that can be realized in G(R) then the character of π(χ, δ), which certainly exists as a distribution, is given on the regular elements of S(R) by the function

ε(G) ∑_{Ω_{2}(G)} \frac{sgnωχ(ωs)e^{δ(ωH−H)}}{Δ(s)} .

It follows that the sets Π_{φ} that have been defined so far are disjoint.

Suppose π is an irreducible quasi-simple representation of G(R) on V that is square-integrable modulo the center. By the previous lemma the restriction of π to G_{0}(R) is the direct sum of finitely many irreducible quasi-simple representations, each of which is clearly square-integrable modulo the center. Let π_{0} be one of them and let π_{0} act on V_{0} ⊆ V. By the theory of the discrete series, G has a Cartan subgroup S over R, so that S(R) ∩ G_{der}(R) is compact and there is a χ and a δ such that π_{0} is π_{0}(χ, δ). By Lemma 3.4 there is a φ such that χ ∈ X_{φ}. If g_{1} …, g_{r} are a set of representatives for G_{0}(R)\G(R) then

V ⊇ ⨁_{i=1}^{r} π(g_{i}^{−1})V_{0}

because the representations h → π(g_{i}hg_{i}^{−1}) are inequivalent. If g_{j}g = h_{j}g_{i} and

v = ⨁ π(g_{i}^{−1})v_{i}

then

π(g)v = ⨁ π(g_{j}^{−1})v′_{j}

with v′_{j} = π(h_{j})v_{i}. Mapping v to the function on G(R) whose value at hg_{i}, h ∈ G_{0}(R) is π(h)v_{i} we obtain an infinitesimal equivalence of π with π(χ, δ). This shows at least that (v) will be a consequence of (i) and that the union of the sets Π_{φ} will contain all classes that are square-integrable modulo the center.

Suppose η : H → G has abelian kernel and cokernel and φ′ = η^{∧}(φ) where φ ∈ Φ(G) and φ(W_{R}) is contained in no proper parabolic subgroup of G^{∧}. Then φ′(W_{R}) is contained in no proper parabolic subgroup of H^{∧}. It follows from the preceding lemma that the restriction of any irreducible quasi-simple representations π of G(R) to H(R) is, infinitesimally, the direct sum ⨁ π_{i} of finitely many irreducible quasisimple representations of H(R), for the map H_{der}^{o}(R) → G_{der}^{0}(R) is surjective. If π is square-integrable modulo the center so is each π_{i}. We consider the restriction of π(χ, δ). π(χ, δ) restricted to G_{0}(R) is the sum

⨁_{Ω_{1}\Ω_{2}(G)} π_{0}(ω^{−1}χ, ω^{−1}δ)

and π_{0}(ω^{−1}χ, ω^{−1}δ) restricted to H_{0}(R) is irreducible. It is clearly equal to π_{0}(ω^{−1}χ′, ω^{−1}δ′) if χ′ is the quasi-character s → χ(η(s)) on the inverse image of S(R) in H(R) and if δ′ is the pullback of δ. It is also easy to see that X′_{φ} = {χ′ | χ ∈ χ_{φ}}. Thus π(χ, δ) restricted to H(R) is

⨁_{Ω_{2}(H)\Ω_{2}(G)} π(ω^{−1}χ′, ω^{−1}δ′)

with χ′ ∈ X′_{φ} and condition (iv) is satisfied.

Condition (ii) is clear when the center of G is connected. In the general case it follows from (iv) and the definition of χ_{φ}. Condition (iii) is clear when G_{der} is simply connected. In the general case it follows from (iv) and the definition of π_{α}.

If the quasi-simple irreducible representation π is square-integrable modulo the center and if ζ is the quasi-character of G_{rad}(R) defined by

π(z) = ζ(z)I z ∈ G_{rad}(R),

then π is tempered if and only if ζ is a character. This is so if and only if ⟨μ, H⟩ is purely imaginary whenever H ∈ L^{∧} × C satisfies σH = \overline{H} and ⟨α, H⟩ = 0 for all α. On the other hand, if φ(w) = a(w) × w then {a(w) | w ∈ W_{R}} is relatively compact if and only if {a(z) | z ∈ C^{×}} is. This is so if and only if

z = e^{x} → λ^{∧}(a(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨\overline{σ}μ,λ^{∧}⟩} = e^{⟨μ,zλ^{∧}+\overline{x} \overline{σ}λ^{∧}⟩}

is a character of C^{×} for each λ^{∧}, that is, if and only if ⟨μ, H⟩ is purely imaginary when H ∈ L^{∧} ⊗ C satisfies \overline{σ}H = \overline{H}. Any such H is a sum of terms of the form xλ^{∧} + \overline{x} \overline{σ}λ^{∧} where λ^{∧} is either a root α^{∧} or satisfies ⟨α, λ^{∧}⟩ = 0 for all α. If λ^{∧} = α^{∧} then

xλ^{∧} + \overline{x} \overline{σ}λ^{∧} = (x − \overline{x})α^{∧}.

Since ⟨μ, α^{∧}⟩ is real, ⟨μ, xλ^{∧}+\overline{x} \overline{σ}λ^{∧}⟩ is purely imaginary. If ⟨α, λ^{∧}⟩ = 0 for all α then \overline{σ}λ^{∧} = σλ^{∧}. Condition (vi) is now clear.

Before completing the definition of the sets π_{φ}, we remind ourselves of some properties of induced representations. Suppose P is a parabolic subgroup of G over R, N its unipotent radical, and M = P/M. Suppose ρ is an irreducible quasi-simple representation of M(R) on a Banach space V. Lifting, we may also treat ρ as a representation of P (R). If p ∈ P (R) let δ_{P} (p) be the square root of the absolute value of the determinant of the restriction of Adρ to the Lie algebra of N. Let I(V) be the space of continuous functions on G(R) with values in V that satisfy

φ(pg) = δ_{P} (p)ρ(p)φ(g), p ∈ P (R).

I(V) is a Banach space; let I_{ρ} be the representation of G(R) on it by right translations.

There is a quasi-character ζ_{ρ} of Z_{G}(R) such that I_{ρ}(z) is the scalar ζ_{ρ}(z) when z ∈ Z_{G}(R) and a unique positive real-valued quasi-character ξ_{ρ} of G(R) such that |ζ_{ρ}(z)| = ξ_{ρ}(z) if z ∈ Z_{G}(R). There is also a quasi-character ζ′_{ρ} of Z_{M}(R) such that ρ(z) is the scalar ζ′_{ρ}(z) if z ∈ Z_{M}(R) and a positive quasi-character ξ′_{ρ} of M(R) such that |ζ′_{ρ}(z)| = ξ′_{ρ}(z) if z ∈ Z_{G}(R). ζ_{ρ} is the restriction of ζ′_{ρ} to Z_{G}(R). If π is an irreducible quasi-simple representation of G(R) we may also define ξ′_{π} and ζ′_{π}.

Suppose ρ^{∗} is a quasi-simple irreducible representation of M(R) on V^{∗} and there is an M(R)-invariant bilinear form

(v, v^{∗}) → ⟨v, v^{∗}⟩ ∈ C

on V × V^{∗}. We may introduce I(V^{∗}), I_{ρ}_{∗}, and the bilinear form

⟨φ, ψ⟩ = ∫_{K}⟨φ(k), ψ(k)⟩dk

on I(V) × I(V^{∗}). It is known to be G(R)-invariant. Any K-finite continuous linear form on I(V) is of the form φ → ⟨φ, ψ⟩ for a suitable ψ in I(V^{∗}). We want to investigate the function ⟨I_{ρ}(g)φ, ψ⟩ for K-finite φ and ψ.

Let X be the lattice of rational characters of P and Y the lattice of rational characters of M_{rad}. There is an injection X → Y that leads to isomorphisms X ⊗ R ⟶^{∼} Y ⊗ R, X ⊗ C ⟶^{∼} Y ⊗ C. We identify these two spaces. If D(P) is the set of invariant elements of X ⊗ R, then every λ = ∑ x_{i}λ_{i} in D(P), λ_{i} ∈ X, x_{i} ∈ R defines a positive quasi-character π_{λ} of M(R) by

π_{λ}(g) = ∏_{i} |λ_{i}(g)|^{x_{i}}, g ∈ M(R).

π_{λ} is a representation on C. If P is minimal over R, we take π_{λ}^{∗} to be π_{λ}^{−1} and φ and ψ to be identically 1 on K and set

ϕ_{λ}(g) = ⟨I_{π_{λ}}(g)φ, ψ⟩.

As usual Ξ_{G} is the function ϕ_{0}.

Recall that if π is a quasi-simple, irreducible representation of G(R) on W that is square-integrable modulo the center, and u and v are K-finite vectors in W and its dual, then ⟨π(g)u, v⟩ is bound by a constant times ζ′_{π}(g)Ξ_{G}(g).

We now prove an easy sequence of lemmas. G is a reductive linear group over R. There is a hermitian form on the underlying real vector space that is invariant under K and with respect to which G(R) is selfadjoint. Every g ∈ G(R) is a product g = kh where k ∈ K and h is selfadjoint and positive with respect to the given form. Let l(g) be the norm of logh. We choose an abelian subgroup A of G(R) every element of which is selfadjoint and positive and which is maximal with respect to this property. A is then connected and G(R) = KAK. If P is a given parabolic subgroup over R we may, and do, take A in P (R).

If we choose ψ : G ⟶^{∼} G′ where G′ is a quasi-split and ψ^{−1}σ(ψ) is inner, if B′ and T′ have the usual significance, and if P′ = ψ(P) contains B′ and ψ(A) ⊆ T′ as we may suppose, then X ⊗ R may be regarded as a subspace of L ⊗ R. Let D^{+}(P) be the set of λ ∈ D(P) such that ⟨λ, α^{∧}⟩ > 0 if α is a root of T′ in N′ = ψ(N). Let \overline{D}^{+}(P) be its closure. If P is minimal let A^{+}(P) be the set of α in A such that α(a) ≥ 1 if α is a root of T′ in N′.

LEMMA 3.6. Let P be minimal over R. There is an integer d and a constant c such that

ϕ_{λ}(a) ≤ cπ_{λ}(a)δ_{P}^{−1}(a)(1 + l(a))^{d}

if a ∈ A^{+}(P), λ ∈ \overline{D}^{+}(P).

The group G(R) = P (R)K. Write g = p(g)k(g). p(g) is not uniquely determined but π_{λ}(p(g)) and δ_{P} (p(g)) are and

ϕ_{λ}(a) = ∫_{K} δ_{P} (p(ka))π_{λ}(p(ka))dk.

By Lemma 3.3.2.3 of [16]

π_{λ}(p(ka) ≤ π_{λ}(a).

Thus

ϕ_{λ}(a) ≤ π_{λ}(a)ϕ_{0}(a) ≤ cπ_{λ}(a)δ_{P}^{−1}(a)(1 + l(a))^{d}.

The last inequality is a consequence of Theorem 8.3.7.4 of [16].

LEMMA 3.7. For each λ ∈ \overline{D}^{+}(P) there is a positive constant c(λ) such that

ϕ_{λ}(a) ≥ c(λ)π_{λ}(a)δ_{P}^{−1}(a)

for all a ∈ A^{+}(P).

To prove this we remind ourselves of an integration formula (cf. [4]). Let P be for the moment any parabolic subgroup over R. Let \overline{N} be the unipotent radical of a parabolic subgroup over R opposed to P.

G(R) is again P (R)K and we write g = p(g)k(g). If f is any continuous function on K ∩ p(R)\K then, with a suitable choice of the Haar measure on \overline{N}(R),

(3.1)  ∫_{K} f(k)dk = ∫_{\overline{N}(R)} δ_{P}^{2} (p(\overline{n}))f(k(\overline{n}))d\overline{n}

Take P to be minimal over R, take

f(k) = δ_{P} (p(ka))π_{λ}(p(ka)),

and write p(\overline{n}) = p_{1}, k(\overline{n}) = k_{1}. Then

k_{1}a = p_{1}^{−1}aa^{−1}\overline{n}a

and

δ_{P} (p(k_{1}a))π_{λ}(p(k_{1}a))

equals

\{δ_{P}^{−1}(p_{1})π_{λ}^{−1}(p_{1}))\}\{δ_{P} (a)π_{λ}(a))\}\{δ_{P} (p(a^{−1}\overline{n}a))π_{λ}(p(a^{−1}\overline{n}a))\}.

Consequently

ϕ_{λ}(a) = δ_{P} (a)π_{λ}(a) ∫_{\overline{N}(R)}\{δ_{P}^{−1}(p_{1})π_{λ}^{−1}(p_{1})\}\{δ_{ρ}(p(a^{−1}\overline{n}a))π_{λ}(p(a^{−1}\overline{n}a))\}d\overline{n}.

Substitute a\overline{n}a^{−1} for \overline{n} to obtain

δ_{P}^{−1}(a)π_{λ}(a) ∫_{\overline{N}(R)}\{δ_{P}^{−1}(p(a\overline{n}a^{−1}))π_{λ}^{−1}(p(a\overline{n}a^{−1}))\}\{δ_{P} (p(\overline{n}))π_{λ}(p(\overline{n}))π_{λ}(p(\overline{n}))\}d\overline{n}.

All we have to do is show that for a given λ ∈ \overline{D}^{+}(P) the integral is bounded below by a positive constant as a varies over A^{+}(P). If U is a relatively compact subset of \overline{N}(R) so is

⋃_{a∈A^{+}(P)} aUa^{−1}.

Since the integrand is continuous and positive, the required estimate is certainly valid.

We shall eventually have to make use of a well-known result of Bhanu-Murty-Gindikin-Karpelevich. If P_{1} is a parabolic subgroup of G over R that contains the minimal P we may set P′_{1} = ψ(P_{1}), N′_{1} = ψ(N_{1}). Suppose \overline{P}_{1} is opposed to P and \overline{N}_{1} is its unipotent radical. If ⟨λ, α^{∧}⟩ > 0 whenever α is a root of T′ in N′_{1} then

(3.2)  ∫_{\overline{N}_{1}(R)} δ_{P} (p(\overline{n}))π_{λ}(p(\overline{n}))d\overline{n} < ∞.

If P and P_{0} are two parabolic subgroups of G over R and P ⊆ P_{0} then D(P) ⊆ D(P_{0}). Also if ξ is any positive quasi-character of G(R) there is a λ in D(G) such that ξ = π_{λ}.

LEMMA 3.8. Suppose P is a parabolic subgroup of G over R. Suppose ρ and ρ^{∗} are two irreducible quasisimple representations of M(R) on V and V^{∗} respectively. Suppose that there is a nontrivial M(R)-invariant pairing V × V^{∗} → C. Let K′ be the projection of K ∩ P (R) on M(R) and suppose that for any two K′-finite vectors v and v^{∗} there is a constant c such that

|⟨ρ(m)v, v^{∗}⟩| ≤ cξ′_{ρ}(m)Ξ_{M}(m), m ∈ M(R).

Suppose ξ′_{ρ} = π_{λ} with λ ∈ \overline{D}^{+}(P). If P contains the minimal P_{0} then λ ∈ \overline{D}^{+}(P_{0}) and for any two K-finite φ and ψ in I(V) and I(V^{∗}) there is a constant c such that

|⟨I_{ρ}(g)φ, ψ⟩| ≤ cϕ_{λ}(g), g ∈ G(R).

As usual we suppose that ψ(P) = P′ and ψ(P′_{0}) contain B′. If α is a root of T′ in N′_{0} that is also a root in n′ then ⟨λ, α^{∧}⟩ ≥ 0. If α is a root in N′_{0} but not in N′ then ⟨λ, α^{∧}⟩ = 0. Consequently λ ∈ \overline{D}^{+}(P_{0}).

If k ∈ K, g ∈ G(R) write kg = pk_{1}, p ∈ P (R), k_{1} ∈ K, and let m be the projection of p on M(R). Then

⟨I_{ρ}(g)φ, ψ⟩ = ∫_{K}⟨φ(kg), ψ(k)⟩dk = ∫_{K}⟨δ_{P} (p)ρ(p)φ(k_{1}), ψ(k)⟩dk.

There are functions φ_{i} in I(V), functions φ_{j} in I(V^{∗}), and continuous functions a_{i}, b_{j} on K such that

φ(hk) = ∑ a_{i}(k)φ_{i}(h), ψ(hk) = ∑ b_{j}(k)ψ_{j}(h)

for h ∈ G(R) and such that φ_{i}(1), ψ_{j}(1) are K′-finite. Then

⟨δ_{P} (p)ρ(p)φ(k_{1}), ψ(k_{1}), ψ(k)⟩ = δ_{P} (p) ∑_{ij} a_{k}(k_{1})b_{j}(k)⟨ρ(m)φ_{i}(1), ψ_{j}(1)⟩.

There is therefore a constant c such that

|⟨δ_{P} (p)ρ(p)φ(k_{1}), ψ(k)⟩| ≤ cδ_{P} (p)π_{λ}(m)Ξ_{M}(m).

We may lift M to a Levi factor of P, chosen so that M(R) is selfadjoint with respect to the given hermitian form. Then K ∩ M(R) is a maximal compact subgroup of M(R) and K ∩ M(R) = K ∩ P (R). The function φ_{λ}(g) is given by

∫_{K} δ_{P_{0}}(p_{0}(kg))π_{λ}(p_{0}(kg))dk = ∫_{K} {∫_{K∩M(R)} δ_{P_{0}}(p_{0}(ukg))π_{λ}(p_{0}(ukg))du}dk.

Set p(kg) = p = nm, n ∈ N(R); then

ukg = unu^{−1}umk_{1}

and unu^{−1} ∈ N(R) ⊆ N_{0}(R). P′_{0} = P_{0} ∩ M is minimal parabolic subgroup of M over R. Write um = p_{0}k_{0}, p_{0} ∈ P′_{0}(R), P_{0} ∈ K ∩ M(R). Then

δ_{P_{0}}(p_{0}(ukg)) = δ_{P} (m)δ_{P′_{0}}(p_{0}).

Because λ ∈ D(P)

π_{λ}(p_{0}(ukg)) = π_{λ}(m).

Thus

ϕ_{λ}(g) = ∫_{K} δ_{P} (m)π_{λ}(m){∫_{K∩M(R)} δ_{P′_{0}}(p_{0})du}dk.

The right side is

∫_{K} δ_{P} (m)π_{λ}(m)Ξ_{M}(m)dk.

Since δ_{P} (p) = δ_{P} (m), the lemma follows.

COROLLARY 3.9. Assume in addition that λ ∈ D(G) ⊆ D(P). Then

|⟨I_{ρ}(g)φ, ψ⟩| ≤ cπ_{λ}(g)Ξ_{G}(g).

One has only to observe that when λ ∈ D(G)

ϕ_{λ}(g) = π_{λ}(g)Ξ_{G}(g).

We shall have to make use of some results from [6] that are considerably more serious than those of the preceeding lemmas. We recapitulate them in the form we require. Two conjugacy classes 𝔭 and 𝔭′ of parabolic subgroups of G over R are said to be associate if we can find P ∈ 𝔭, P′ ∈ 𝔭′ such that P and P′ have a common Levi factor. Given P and the Levi factor M, for which M(R) is selfadjoint, we may, with no loss of generality, assume that it is the common factor, for we may replace P′ by a conjugate.

If ρ is a quasi-simple irreducible representation of M(R) we may define I_{ρ} with respect to P or to P′. To distinguish the two possibilities we write I_{ρ}^{P}, I_{ρ}^{P′}. To apply the results of [6] we take ρ to be square-integrable modulo the center. It then satisfies the conditions of Lemma 3.8. In fact, we may suppose, since it is only the infinitesimal equivalence class of ρ and I_{ρ} that interests us, that V is a Hilbert space and that π_{λ}^{−1} ⊗ ρ is unitary. Then we take V^{∗} to be the dual space to V and ρ^{∗} to be the representation contragredient to ρ. If (u, v) is the inner product on V then

(φ, ψ) = ∫_{K}(φ(k), ψ(k))dk

is an inner product on I(V). If we assume in addition, as we must, that ρ satisfies the condition of the corollary, then π_{λ}^{−1}(g)I_{ρ}(g) is unitary with respect to this inner product.

For a general quasi-simple irreducible ρ, the elements of ξ operate on the infinitely differentiable vectors in I(V) as scalars. Moreover the restriction of I_{ρ}^{P} to K contains any irreducible representation of K with finite multiplicity. Exploiting, for example, the fact that the characters of irreducible quasi-simple representations of G^{o}(R) are functions, one sees that I_{ρ}^{P} admits a finite composition series. Our present stronger assumptions on ρ, which imply the existence of an inner product on I(V) with respect to which the operators I_{ρ}(g) differ by a scalar from a unitary matrix, allow us to conclude that I_{ρ} is infinitesimally equivalent to the direct sum of finitely many irreducible quasi-simple representations. From Lemma 8 and Theorem 3 of [6], we conclude that if I_{ρ}^{P} and I_{ρ′}^{P′} have an irreducible constituent in common then the classes of P and P′ are associate.

If P and P′ have the common Levi factor M, then, by computing the character, one sees that I_{ρ}^{P} and I_{ρ′}^{P} are infinitesimally equivalent ([6], ∮ 11). By Lemma 12 of [6], the representations I_{ρ}^{P} and I_{ρ′}^{P} are infinitesimally equivalent if and only if there is an h in the normalizer of M in G(R) such that ρ′ and m → ρ(hmh^{−1}) are infinitesimally equivalent. What does not appear so clearly in [6] is that if I_{ρ}^{P} and I_{ρ′}^{P} have an irreducible constituent in common, then there is an h in the normalizer of M in G(R) such that ρ′ and m → ρ(hmh^{−1}) are infinitesimally equivalent.

This is an important point. We shall return to it after some considerations that are, unfortunately, only implicit in [6]. We take up once again the assumptions of Lemma 3.8. Suppose ψ ∈ V^{∗} is K ∩ M(R) finite. If φ ∈ I(V) is K-finite then φ(k) ∈ V is K ∩ M(R) finite for all k ∈ K and {φ(k)|k ∈ K} spans a finite-dimensional subspace of V. There is therefore a constant c such that

|⟨ρ(m)φ(k), ψ⟩\ ≤ cπ_{λ}(m)Ξ_{M}(m)

for all m ∈ M(R) and all k ∈ K. Suppose φ ∈ I(V), ψ ∈ V^{∗}, and

(3.3)  sup_{k,m} \frac{|⟨ρ(m)φ(k), ψ⟩|}{π_{λ}(m)Ξ_{M}(m)} = ∥φ∥_{ψ} < ∞.

If U is a compact subset of G(R) and p(kg) = n(kg)m(kg) then

{m(kg)|k ∈ K, g ∈ U}

is relatively compact. Set m_{1} = m(kg), k_{1} = k(kg), and let m_{2} ∈ M(R); then

|⟨ρ(m)φ(kg), ρ(m_{2})ψ⟩| = δ_{P} (m_{1})|⟨ρ(m_{2}^{−1}mm_{1})φ(k_{1}), ψ⟩|

which is bounded by

cπ_{λ}(m_{2}^{−1}mm_{1})Ξ_{M}(m_{2}^{−1}mm_{1}) ≤ c′π_{λ}(m)Ξ_{M}(m).

For the last inequality see Proposition 8.3.7.2 of [16].

It follows easily from these considerations that if f ∈ C_{c}^{∞}(G(R)), if ∥φ′∥_{ψ} < ∞, and

φ = I_{ρ}(f)φ′ = ∫_{G(R)} f(g)I_{ρ}(g)φ′dg

then

(3.4)  lim_{g→h} ∥I_{ρ}(g)φ − I_{ρ}(h)φ∥_{ψ} = 0.

Notice in particular that if φ is K-finite there is an f such that

φ = I_{ρ}(f)φ.

If ψ ∈ V^{∗} let I(V, ψ) be the set of all φ ∈ I(V) satisfying (3.3) and (3.4). If v is a compactly supported measure on M(R) and

ψ′ = ∫_{M(R)} ρ(m)ψdv

then I(V, ψ′) ⊇ I(V, ψ). In particular I(V, ψ) is the same for all nonzero K ∩ M(R) finite vectors ψ. If I(V, ψ) contains the K-finite vectors then the restriction of I_{ρ} to I(V, ψ) is infinitesimally equivalent to ψ.

LEMMA 3.10. Suppose ρ satisfies the conditions of Lemma 3.8 with λ ∈ D(P)^{+}. Suppose M is a Levi factor of P with M(R) selfadjoint and \overline{P} ∩ P = M. If \overline{N} is the unipotent radical of \overline{P}, ψ ∈ V^{∗}, and φ ∈ I(V, ψ) then

∫_{\overline{N}(R)}⟨φ(\overline{n}g), ψ⟩d\overline{n}

is absolutely convergent.

We may take g = 1. Write \overline{n} = nmk, n ∈ N(R), m ∈ M(R), k ∈ K. Then
|⟨φ(\overline{n}), ψ⟩| = δ_{P} (m)|⟨ρ(m)φ(k), ψ⟩| ≤ cδ_{P} (m)π_{λ}(m)Ξ_{M}(m).

We have seen that if P_{0} ⊆ P is minimal then

∫_{\overline{N}(R)} δ_{P}_{0}(p_{0}(\overline{n}))π_{λ}(p_{0}(\overline{n}))d\overline{n}

is infinite; it equals

∫_{\overline{N}(R)} ∫_{K∩M(R)} δ_{P_{0}}(p_{0}(u\overline{n}u^{−1})π_{λ}(p_{0}(u\overline{n}u^{−1}))d\overline{n}.

Since p_{0}(u\overline{n}u^{−1}) = p_{0}(u\overline{n}) we may proceed as in the proof of Lemma 3.8 to see that this double integral equals

∫_{N(R)} δ_{P} (m)π_{λ}(m)Ξ_{M}(m)d\overline{n}.

The lemma follows.

We set

∫_{\overline{N}(R)}⟨φ(\overline{n}g), ψ⟩d\overline{n} = M(φ, ψ; g).

It is clear that

M(φ, ψ; gh) = M(I_{ρ}(h)φ, ψ; g)

and that

M(φ, ρ(m)ψ, g) = δ_{P}^{−2}(m)M(φ, ψ; m^{−1}g).

Let V_{0}^{∗} be the space of vectors ψ in V^{∗} for which I(V, ψ) contains the K-finite vectors. V_{0}^{∗} is invariant under M(R). If φ is K-finite

ψ → M(φ, ψ, 1)

is a K ∩ M(R)-finite linear form on V_{0}^{∗}. There is therefore a K ∩ M(R)-finite vector M(φ) in V such that

M(φ, ψ, 1) = ⟨M(φ), ψ⟩

for ψ ∈ V_{0}^{∗}.

The map p × \overline{n} → p\overline{n} of P (R) × \overline{N}(R) into G(R) is a diffeomorphism of P (R) × \overline{N}(R) with an open subset of G(R). If f is an infinitely differentiable complex-valued function on \overline{N}(R) with compact support X and v ∈ V is K-finite, define φ ∈ I(V) by

φ(p\overline{n}) = f(\overline{n})δ_{P} (p)ρ(p)v.

The set

Y = {m ∈ M(R)|N(R)mK ∩ X ≠ ∅}

is compact. If m ∈ M(R) and mk = n_{1}m_{1}\overline{n}, n_{1} ∈ N(R), m_{1} ∈ M(R), \overline{n} ∈ X then m = m_{1}m_{2} with m_{2} ∈ Y. Thus if ψ is K ∩ M(R)-finite

|⟨ρ(m)φ(k), ψ⟩| ≤ cπ_{λ}(m_{1})Ξ_{M}(m_{1}) ≤ c′π_{λ}(m)Ξ_{M}(m).

Given ψ we can clearly choose v and f such that M(φ, ψ, 1) ≠ 0. If d in C_{c}^{∞}(g(R)) is a sufficiently close approximation to the delta-function and φ′ = I_{ρ}(d)φ then M(φ′, ψ, 1) is also not zero. Since φ′ ∈ I(V, ψ) we have the following lemma.

LEMMA 3.11. If ψ ∈ V^{∗} is K ∩ M(R)-finite there is a φ ∈ I(V, ψ) such that

M(φ, ψ, 1) ≠ 0.

If M(R) is selfadjoint and A ⊆ P (R) then A ⊆ M(R). Let A(P) be the centralizer of M(R) in A and let A^{+}(P) consist of those a in A(P) for which α(a) ≥ 1 if α is a root of T′ in N′. We say that a → ∞ in A^{+}(P) if α(a) → ∞ for all such α.

LEMMA 3.12. Suppose that φ ∈ I(V) and ψ ∈ I(V^{∗}) are K-finite. If m ∈ M(R) is fixed, then

⟨I_{ρ}(am)φ, ψ⟩ = δ_{P}^{−1}(a){ζ′_{ρ}(a)M(φ, ψ(1), m) + o(π_{λ}(a))}

as a → ∞ in A^{+}(P).

Since |ζ′_{ρ}(a)| = π_{λ}(a) the error term is smaller than the principal term if M(φ, ψ(1), m) ≠ 0.

Replacing φ by I_{ρ}(m), we suppose m = 1. φ may no longer be K-finite.

⟨I_{ρ}(a)φ, ψ⟩ = ∫_{K}⟨φ(ka), ψ(k)⟩dk.

The integrand is clearly a function on K ∩ P (R)\K. Choose P and \overline{N} as in the previous lemma and write \overline{n} = nmk, n ∈ N(R), m ∈ M(R), k ∈ K. Applying (3.1) we see that the integral is equal to

∫_{\overline{N}(R)} δ_{P}^{2} (m)⟨φ(ka), ψ(k)⟩, d\overline{n}.

Let a^{−1}\overline{n}a = n_{1}m_{1}k_{1}. Since ka = m^{−1}n^{−1}\overline{n}a = m^{−1}n^{−1}aa^{−1}\overline{n}a the intergrand is

δ_{P} (m)δ_{P} (a)δ_{P} (m_{1})⟨ρ(m^{−1}am_{1})φ(k_{1}), ψ(k)⟩.

If we substitute a\overline{n}a^{−1} for \overline{n} so that \overline{n} = n_{1}m_{1}k_{1}, a\overline{n}a^{−1} = nmk the integral becomes

δ_{P}^{−1}(a)ζ′_{ρ}(a) ∫_{\overline{N}(R)} δ_{P} (m)δ_{P} (m_{1})⟨ρ(m^{−1}m_{1})φ(k_{1}), ψ(k)⟩d\overline{n}.

All we have to do is show that

lim_{a→∞} ∫_{\overline{N}(R)} δ_{P} (m)δ_{P} (m_{1})⟨ρ(m^{−1}m_{1})φ(k_{1}), ψ(k)⟩d\overline{n} = M(φ, ψ(1), 1).

In a moment we shall show that we may take the limit under the integrals ign. Since a\overline{n}a^{−1} → 1, we may suppose that n → 1, m → 1, k → 1. The integrand approaches

δ_{P} (m_{1})⟨ρ(m_{1})φ(k_{1}), ψ(1)⟩ = ⟨φ(\overline{n}), ψ(1)⟩

and the lemma follows.

The integral is dominated by a constant times

δ_{P} (m)δ_{P} (m_{1})π_{λ}(m^{−1}m_{1})Ξ_{M}(m^{−1}m_{1}).

Choose P_{0} ⊆ P such that P_{0}(R) ⊇ A and such that P_{0} is minimal over R. As usual let P′_{0} = P_{0} ∩ M. Since

Ξ_{M}(m^{−1}m_{1}) = ∫_{K∩M(R)} δ_{P′_{0}}(p′_{0}(um^{−1}m_{1}))du,

it is enough to show that the integral of

δ_{P} (m)δ_{P} (m_{1})π_{λ}(m^{−1}m_{1})δ_{P′_{0}}(p′_{0}(um^{−1}m_{1}))

is uniformly small on the complement of a large compact set in \overline{N}(R) × (K ∩ M(R)).

Choose a P_{0} ⊆ P that is opposed to P_{0} so that

A ⊆ \overline{P}_{0}(R) ∩ P_{0}(R) = M_{0}(R) ⊆ M(R).

Write an element of \overline{N}_{0}(R) as \overline{n}_{0} = n_{0}m_{0}k_{0}, n_{0} ∈ N_{0}(R), m_{0} ∈ M_{0}(R), k_{0} ∈ K. If f is a function on K ∩ M_{0}(R)\K then

∫_{K} f(k)dk = ∫_{\overline{N}_{0}(R)} δ_{P_{0}}^{2}(m_{0})f(k_{0})d\overline{n}_{0}.

If \overline{N}′_{0} = \overline{N}_{0} ∩ M then \overline{N}_{0}(R) = \overline{N}′_{0}(R)\overline{N}(R). Let \overline{n}_{0} = \overline{n}′_{0}\overline{n}_{1}; let \overline{n}′_{0} = n_{2}m_{2}v, n_{2} ∈ N′_{0}(R), m_{2} ∈ M_{0}(R), k ∈ K ∩ N(R); let v\overline{n}_{1}v^{−1} = n_{3}m_{3}k_{3}, n_{3} ∈ N_{0}(R), m_{3} ∈ M_{0}(R), k_{3} ∈ K. We may suppose

m_{0} = M_{2}M_{3}, k_{0} = K_{3}u.

Then the integral is equal to

∫_{\overline{N}′_{0}(R)} ∫_{\overline{N}(R)} δ_{P_{0}}^{2}(m_{2}m_{3})f(k_{3}v)d\overline{n}_{1}d\overline{n}′_{0}.

Replacing \overline{n}_{1} by a\overline{n}_{1}a^{−1} so that n_{3}m_{3}k_{3} = va\overline{n}_{1}a^{−1}v^{−1} we obtain

δ_{P}^{−2}(a) ∫_{\overline{N}′_{0}(R)} ∫_{\overline{N}(R)} δ_{P_{0}}^{2}(m_{2}m_{3})f(k_{3}v)d\overline{n}_{1}d\overline{n}′_{0}.
On the other hand, if \overline{n} = nmk
∫_{K} f(k)dk = ∫_{\overline{N}(R)} δ_{P}^{2} (m)\{∫_{K∩m(R)} f(uk)du\}d\overline{n}.

Replacing \overline{n} by a\overline{n}a^{−1}, we obtain

δ_{P}^{−2}(a) ∫_{\overline{N}(R)} δ_{P}^{2} (m)_{\{∫K∩M(R)}f(uk)du\}d\overline{n}

where nmk is now a\overline{n}a^{−1}.

Thus

∫_{\overline{N}(R)} δ_{P}^{2} (m)\{∫_{K∩M(R)} f(uk)du\}d\overline{n} = ∫_{N′_{0}(R)} ∫_{N(R)} δ_{P}^{2} (m_{2}m_{3})f(k_{3}v)d\overline{n}_{1}d\overline{n}′_{0}.

If X is a compact set in \overline{N}′_{0}(R) × \overline{N}(R) = \overline{N}_{0}(R) with complement 𝒞X, then

Y = \{\overline{n} ∈ \overline{N}(R)|\overline{n} ∈ P (R)X\}

is also compact. If f is positive

∫_{𝒞Y} δ_{P}^{2} (m)\{∫_{K∩M(R)} f(uk)du\}d\overline{n} ≤ ∫_{𝒞X} δ_{P}^{2} (m_{2}m_{3})f(k_{3}v)d\overline{n}_{1}d\overline{n}′_{0}.

Take f to be the function

δ_{P}^{−1}(a)π_{λ}^{−1}(a)π_{λ}(p_{0}(ka))δ_{P}_{0}(p_{0}(ka)).

If a\overline{n}a^{−1} = nmk, then uka = um^{−1}n^{−1}a\overline{n} and

δ_{p}^{2}(m)f(uk) = δ_{P} (m)δ_{P} (m_{1})π_{λ}(m^{−1}m_{1})δ_{P′_{0}}(p′_{0}(um^{−1}am_{1}))

if \overline{n} = n_{1}m_{1}k_{1}. Observe that

δ_{P} (a) = δ_{P}_{0}(a)

and that, more generally,

δ_{P}_{0}(m) = δ_{P} (m)δ_{P′_{0}}(p′_{0}(m))

for all m in M(R).

To finish the proof of the lemma, we show that

δ_{P}^{2} (m_{2}m_{3})f(k_{3}v)

is dominated on \overline{N}_{0}(R) by an integrable function that is independent of a. Let v\overline{n}_{1}v^{−1} = n_{4}m_{4}k_{4}, n_{4} ∈ N_{0}(R), m_{4} ∈ M_{0}(R), k_{4} ∈ K. Since k_{3}va = m_{0}^{−1}n_{0}^{−1}a\overline{n}_{0} and m_{0} = m_{2}m_{3},

δ_{P_{0}}^{2}(m_{2}m_{3})f(k_{3}v) = δ_{P′_{0}}^{2}(m_{2})δ_{P_{0}}(m_{3}m_{4})π_{λ}(m_{3}^{−1}m_{4})

m_{2} does not depend on \overline{n}_{1}. Consider the function

δ_{P_{0}}(m_{3}m_{4})π_{λ}(m_{3}^{−1}m_{4})

on \overline{N}(R). Replacing \overline{n}_{1} by v^{−1}\overline{n}_{1}v we may, for the present purposes, suppose that v = 1. There is a δ in D^{+}(P_{0}) such that

δ_{P_{0}}(m) = π_{δ}(m), m ∈ M_{0}(R).

There is a β, 1 > β > 0, such that δ − βλ also lies in D^{+}(P_{0}). Let α = 1 − β. By Lemma 13 of [4],

π_{δ−β_{λ}}(m_{3}) ≤ 1, π_{αλ}(m_{3}^{−1}m_{4}) ≤ 1.

By the formula of Bhanu-Murty-Gindikin-Karpelevich [3], the function

π_{δ+βλ}(m_{4})

is integrable on \overline{N}(R). It does not depend on a. Since the transformation \overline{n}_{1} → v^{−1}\overline{n}_{1}v is unimodular, it remains only to observe that, by the same formula,

∫_{\overline{N}′_{0}(R)} δ_{P′_{0}}^{2}(m_{2})d\overline{n}′_{0} < ∞.

We still suppose that λ ∈ D(P^{+}). The space I(V, ψ) is the same for all nonzero K ∩ M(R)-finite ψ. Denote it by I_{0}(V) and provide it with the norm

∥φ∥_{ψ} + sup_{k∈K} ∥φ(k)∥.

This norm depends in no essential way on ψ. The subspace

I_{1}(V) = {φ ∈ I_{0}(V)|M(φ, ψ, g) = 0 for all g ∈ G(R)}

is also independent of ψ. It is closed and G(R)-invariant. The quotient J(V) = I_{0}(V)/I_{1}(V) is not zero. Let J_{ρ} be the representation of G(R) on it. When we want to indicate the presence of P we write J_{ρ}^{P} instead of J_{ρ}.

LEMMA 3.13. The representation J_{ρ} is irreducible.

Suppose I_{2}(V) is a closed invariant subspace of I_{0}(V) and I_{1}(V) ⊄ I_{2}(V) ⊄ I_{0}(V). If v^{∗} ∈ V^{∗} is K ∩ M(R)-finite then the function

φ → M(φ, v^{∗}, 1)

on I_{0}(V) is continuous. If it vanishes on all K-finite functions in I_{2}(V) it vanishes identically on I_{2}(V). Then

0 = M(I_{ρ}(g)φ, v^{∗}, 1) = M(φ, v^{∗}, g)

for all φ ∈ I_{2}(V). This is impossible unless V^{∗} = 0. On the other hand, there is a K-finite function ψ in I(V^{∗}) that is not zero but is orthogonal to I_{2}(V). Then ⟨I_{ρ}(g)φ, I_{ρ}^{∗}(k)ψ⟩ = 0 for all g ∈ G(R) and all k ∈ K. If φ in I_{2}(V) is K-finite we may apply Lemma 3.12 to see that

M(φ, ψ(k), 1) = 0

for all k. This in turn implies that ψ(k) = 0 for all k, which is a contradiction. The lemma follows.

LEMMA 3.14 (a) Suppose ρ and ρ′ satisfy the conditions of Lemma 3.8 with respect to P and P′ respectively. Suppose λ ∈ D^{+}(P) and λ′ ∈ D^{+}(P′). If J_{ρ}^{P} and J_{ρ′}^{P′} are infinitesimally equivalent there is an h ∈ G(R) such that P′ = hph^{−1}, M′ = hMh^{−1}, and such that m → p′(hmh^{−1}) is infinitesimally equivalent to ρ.

(b) Suppose ρ satisfies the conditions of Lemma 3.8 with respect to P and λ ∈ D^{+}(P). Suppose ρ′ satisfies the conditions of Corollary 3.9 with respect to P′. If P ≠ G, then J_{ρ}^{P} is infinitesimally equivalent to no constituent of J_{ρ′}^{P′}.

We may certainly suppose that P and P′ both contain P_{0} minimal over R, that A ⊆ P_{0}(R), and that M(R) and M′(R) are selfadjoint.

If π and π′ are two irreducible quasi-simple representations of G(R) on W and W′, respectively, then π and π′ are infinitesimally equivalent if and only if for any K-finite vector w ∈ W and any K-finite linear form f on W there are a K-finite w′ ∈ W′ and a K-finite linear form f′ on W′ such that

f(π(g)w) = f′(π′(g)w′)

for all g ∈ G(R). If S(π) is the set of all ν ∈ D(P_{0}) such that for any K-finite f and w there is a constant c such that

|f(π(a)w)| ≤ cδ_{P_{0}}^{−1}(a)π_{ν}(a)

on A^{+}(P_{0}) then

S(π) = S(π′)

S(π) is a convex set. We introduce the Killing form B(μ, ν) on L ⊗ R. It is positive semidefinite. If S(π) is not empty there is a unique point v(π) in its closure such that

b(v(π), v(π)) = inf{B(ν, ν)|ν ∈ S(π)}.
If π is J_{ρ}^{P} there is a K-finite φ in I_{0}(V) and a K-finite ψ in I(V^{∗}) such that
f(π(g)w) = ⟨I_{ρ}(g)φ, ψ⟩

for all g. By Lemmas 3.6 and 3.8, the closure of S(π) contains λ.

We may choose a K-finite ψ in I(V^{∗}) such that ψ is orthogonal to I_{1}(V) and such that ψ(1) ≠ 0. We may also choose a K-finite φ in I_{0}(V) such that M(φ, ψ, 1) ≠ 0. Applying Lemma 3.12 to this pair we see that if ν = λ + μ belongs to S(π) then μ(a) ≥ 1 if a ∈ A^{+}(P). Since λ ∈ D^{+}(P), B(λ, μ) is then nonnegative and

B(ν, ν) = B(λ, λ) + 2B(λ, μ) + B(μ, μ) ≥ B(λ, λ).

Thus ν(π) = λ.

If ρ′ satisfies the conditions of Lemma 3.9 and π′ is a constituent of I_{ρ′}^{P′} then we can find φ′ and ψ′ such that

f′(π′(g)w′) = ⟨I_{ρ′}(g)φ′, ψ′⟩.

It follows readily from Lemma 3.6 and Corollary 3.9 that λ′ lies in the closure of S(π′). Since λ′ ∈ D(G) = D^{+}(G), B(λ′, λ′) = 0 and ν(π′) = λ′.

If P and P′ contain P_{0}, D^{+}(P) and D^{+}(P′) are disjoint unless P = P′. This gives the second part of the lemma and half of the first. We now suppose that P = P′, M = M′ and show that if J_{ρ}^{P} and J_{ρ′}^{P′} are infinitesimally equivalent then so are ρ and ρ′.

Let ψ in I(V^{∗}), with ψ(1) ≠ 0, be K-finite and orthogonal to I_{1}(V). Let φ lie in I_{0}(V) but not in I_{1}(V) and be K-finite. Then there exist φ′ and ψ′ that satisfy analogous conditions with respect to ρ′ such that

⟨I_{ρ}(g)φ, ψ⟩ = ⟨I_{ρ′}(g)φ′, ψ′⟩.

Applying Lemma 3.12 we see that

⟨ρ(m)M(φ), ψ(1)⟩ = ⟨M(φ), ρ^{∗}(m^{−1})ψ(1)⟩
= M(φ, ρ^{∗}(m^{−1})ψ(1)1) = M(φ, ψ(1)m)

is equal to

⟨ρ′(m)M(φ′), ψ′(1)⟩.

Since M(φ) ≠ 0, it follows that ρ and ρ′ are infinitesimally equivalent.

There is a point mentioned earlier that remains to be settled. We have to show that if ρ and ρ′ satisfy the conditions of Lemma 3.9 with respect to P and I_{ρ}^{P} and I_{ρ′}^{P} contain infinitesimally equivalent irreducible constituents then there is an h in the normalizer of M in G(R) such that ρ′ and m → ρ(hmh^{−1}) are infinitesimally equivalent.

We have defined the quasi-character π_{ν} on M(R) for ν ∈ D(P). The same formula

π_{ν}(g) = ∏_{i} |λ_{i}(g)|^{x_{i}}

if x = ∑ x_{i}λ_{i} serves to define it for ν ∈ D(P) ⊗ C. Set ρ_{ν} = π_{ν} ⊗ ρ. All the representations ρ_{ν} act on the same space. Take ν in iD(P) and consider the functions

f^{ν}(g) = ⟨I_{P_{ν}} (g)φ, ψ⟩

where φ ∈ I(V) and ψ ∈ I(V^{∗}) are K-finite. Let h run over a set of representatives for the normalizer of M in G(R) modulo M(R). If v does not lie in a certain finite set of hyperplanes the quasi-characters ζ_{h}^{ν} : a → ρ_{ν}(hah^{−1}) of A(P) are distinct and if f_{P} is defined as in [6] then, by Theorem 5 of that paper

f_{P}^{ν} (m) = ∑_{h} θ_{h}^{ν}(m)

where

θ_{h}^{ν}(am) = ζ_{h}^{ν}(a)θ_{h}^{ν}(m), a ∈ A(P).

By Lemma 3.12 and by Lemma 9 of [6], θ_{1}^{ν}(m) has the form

θ_{1}^{ν}(m) = ⟨ρ_{ν}(m)M_{ν}(φ), ψ(1)⟩

where M_{ν}(φ) lies in V. M_{ν}(φ) is a meromorphic function of ν in a neighborhood of iD(P) and its singularity can be killed by a product of linear factors. Let ρ_{ν}^{h} be the representation m → ρ_{ν}(jmh^{−1}). Since I_{ρ_{ν}} and I_{ρ_{ν}^{h}}. It follows that θ_{h}^{ν} is of the form

⟨ρ_{ν}^{h}(m)M_{ν}^{h}(φ), N_{ν}^{h}(ψ)⟩

where M_{ν}^{h}(φ), N_{ν}^{h}(ψ) are K∩M(R)-finite vectors in V and V^{∗}. They too are meromorphic in a neighborhood iD(P) and their singularities can be killed by a product of linear factors.

Let λ_{1}, …, λ_{r} be a basis for D(P). If a = (a_{1}, …, a_{r}) is an r-tuple of non-negative integers set

l_{a}(m) = ∏_{i=1}^{r}{log π_{λ_{i}}(m)}^{a_{i}}.

Let {h} be a set of representatives for the normalizer of M in G(R) modulo the group of those g in the normalizer for which ρ^{g} : m → ρ(gmg^{−}1) is equivalent to ρ. If the reader is willing to admit* that, for a fixed m, f_{P}^{ν} (m) depends continuously on ν ∈ iD(P), he can conclude that

f_{P}^{o} (m) = ∑_{h} ∑_{a} l_{a}(m)⟨ρ^{h}(m)φ_{a}^{h}, ψ_{a}^{h}⟩

where the sum on a is finite and φ_{a}^{h} and ψ_{a}^{h} are K ∩ M(R)-finite vectors in V and V^{∗}.

If M is also a Levi factor of P′ then a similar result holds for f_{P′}^{o}(m). Since not all of the functions f_{P′}^{o}(m) can vanish identically, we conclude that from a nonzero matrix coefficient of any irreducible constituent of I_{ρ}^{P} we can retrieve at least one nonzero matrix coefficient of one of the representations ρ^{h}. This yields the required assertion.

We are now in a position to complete our definition of sets Π_{φ}. Let P^{∧} be minimal among the parabolic subgroups of G^{∧} containing φ(W_{R}). The group G^{∧} may be represented as a semidirect product Gô × W_{R}.

* The author leaves him to struggle with his conscience.

Since C^{×} ⊆ W_{R} acts trivially we may divide out by it to obtain the algebraic group Gô × 𝔤(C/R). The image of P^{∧} is also algebraic and we may use the theorem that any two maximal fully reducible subgroups in an algebraic group are conjugate to conclude that φ(W_{R}) is contained in a Levi factor M^{∧} of P^{∧}. By assumption the class of P lies in 𝔭^{∧}(G). Let P be a parabolic subgroup of G over R whose class corresponds to that of P^{∧}. Since φ(W_{R}) is contained in no proper parabolic subgroup of M^{∧} the earlier definition, together with Lemma 2.4, associates to φ a finite set Π_{φ}(P, P^{∧}) in Π(M).

Suppose φ(W_{R}) is contained minimally in both P^{∧} and \overline{P}^{∧}. Then

(P^{∧} ∩ \overline{P}^{∧})N^{∧} ⊇ (P ô ∩ \overline{P} ô)N^{∧}

and the right side is a parabolic subgroup of Gô ([1], Proposition 4.4). The right side is the connected component of the left. Since the left contains φ(W_{R}) which projects onto W_{R}, it is a parabolic subgroup of G^{∧}. Since it is contained in P^{∧} and contains φ(W_{R}) it is equal to P^{∧}. By the same proposition \overline{P} ô contains a maximal reductive subgroup of P ô. Reversing the roles of the two groups we see that P ô contains a maximal reductive subgroup of \overline{P} ô. Since any two maximal reductive subgroups of P ô or of \overline{P} ô have the same dimension, P ô and \overline{P} ô have a common maximal reductive subgroup. As before, we can divide G^{∧} by C^{×} ⊆ G^{∧} = Gô ×W_{R} to obtain an algebraic group. C^{×} is contained in P^{∧} ∩\overline{P}^{∧} and the quotient of P^{∧} ∩\overline{P}^{∧} by C^{×} is an algebraic group. Take a maximal reductive subgroup in it which contains the image of φ(W_{R}). Let its inverse image in P^{∧} ∩ \overline{P}^{∧} be M^{∧}. M^{∧} contains φ(W_{R}) and therefore projects onto W_{R}. Since [12] any two maximal reductive subgroups of the quotient of P^{∧} ∩ \overline{P}^{∧} by C^{×} are conjugate, M^{∧} contains a Levi factor of P ô and of \overline{P} ô. Thus M^{∧} itself is a Levi factor of P^{∧} and of \overline{P}^{∧}.

Since the set Π_{φ}(P, P^{∧}) does not depend on M^{∧}, we may, for our purposes, fix M^{∧} and let P^{∧}, which does affect Π_{φ}(P, P^{∧}), vary over the parabolic subgroups of G^{∧} with M^{∧} as Levi factor. Since the pair (M, P) together with the set Π_{φ}(P, P^{∧}) is determined only up to conjugacy we may assume that M too is fixed. It will be supposed that M(R) is selfadjoint, although this is not important.

If D(M) is the space introduced in the proof of Lemma 2.5 and M is a Levi factor of P then D(M) = D(P). Let λ = λ_{φ}(P, P^{∧}) in D(M) be defined by the condition that

|ζ′_{ρ}(z)| = π_{λ}(z)

if ρ ∈ Π_{φ}(P, P^{∧}) and z ∈ Z_{M}(R). We observe next that P and P^{∧} may always be so chosen that λ_{φ}(P, P^{∧}) ∈ \overline{D}^{+}(P). In fact, if we vary P and P^{∧} simultaneously as in Lemma 2.5 we may let P vary over all parabolic subgroups of G with M as Levi factor without affecting Π_{φ}(P, P^{∧}) or λ_{φ}(P, P^{∧}). For at least one such P, λ_{φ}(P, P^{∧}) ∈ \overline{D}^{+}(P), the closure of the chamber corresponding to P. From now on we only consider pairs P, P^{∧} for which λ_{φ}(P, P^{∧}) ∈ \overline{D}^{+}(P). There is, moreover, a unique parabolic subgroup P_{1} of G over R such that P_{1} ⊇ M and λ_{φ}(P, P^{∧}) ∈ D^{+}(P_{1}). P_{1} contains P and there is a unique P_{1}^{∧} containing P^{∧} such that P_{1} and P_{1}^{∧} lie in corresponding classes.

We can characterize P_{1}^{∧} in terms of M^{∧} and φ(W_{R}) alone, without reference to P and P^{∧}. For this we shall have to take cognizance of the way M^{∧} is identified with the associate group of M. We recall that we choose an isomorphism ψ of G with a quasi-split group G′, containing B′ and T′, so that ψ^{−1}σ(ψ) is inner and so that ψ(P) ⊇ B′ and M′ = ψ(M) ⊇ T′. We then use G′, B′, and T′ to construct Gô, Bô, T ô and G^{∧} = Gô × W_{R}. Conjugating M^{∧} and P^{∧}, and therefore also φ, but that does not matter, we arrange that P^{∧} ⊇ Bô and that M^{∧} ⊇ T ô. The construction of G^{∧} is such that this M^{∧} can be trivially identified with the associate group of M. We may also suppose that φ(C^{×}) ⊆ T ô × C^{×}.

Now that everything is explicit, let us recall how the restriction of ζ′_{ρ}, ρ ∈ Π_{φ}(P, P^{∧}), to the connected component of M_{rad}(R) is determined by φ. We write φ(w) = a(w)×w with respect to the splitting Gô×W_{R}. As before let \overline{σ} be the action of σ ∈ 𝔤(C/R) on T ô, L, and L^{∧} determined by φ. Choose μ and ν = \overline{σ}μ in L ⊗ C such that

λ^{∧}(a(z)) = z^{⟨μ,λ^{∧}⟩}\overline{z}^{⟨ν,λ^{∧}⟩}

Any s in the connected component of M_{rad}(R) may be written as s = e^{H}, where H = \overline{σ}\overline{H} lies in L^{∧} ⊗ C and ⟨α, H⟩ = 0 if α is a root of T′ in M′. Then

ζ′_{ρ}(s) = e^{⟨μ/2,H+\overline{σ}\overline{H}⟩} = e^{⟨μ,\overline{H}⟩}.

Write μ = μ_{1} + μ_{2} where \overline{σ}μ_{1} = μ_{1}, \overline{σ}μ_{2} = −μ_{2}. Because φ(W_{R}) is contained in no proper parabolic subgroup of M^{∧}, ⟨μ_{1}, α^{∧}⟩ = 0 if α^{∧} is a root of T ô in M ô. Since μ − \overline{σ}μ = 2μ_{2} lies in L,

\overline{⟨μ_{2}, H⟩} = ⟨μ_{2}, \overline{H}⟩ = ⟨\overline{σ}μ_{2}, \overline{σ}\overline{H}⟩ = −⟨μ_{2}, H⟩

and ⟨μ_{2}, H⟩ is purely imaginary. A similar calculation shows that

Re⟨μ_{1}, H⟩ = ⟨Reμ_{1}, H⟩.

Thus

|ζ′_{ρ}(s)| = e^{⟨Reμ_{1},H⟩} = e^{Reμ_{1},H+\overline{H}⟩/2} = ∏_{i} |λ_{i}(s)|^{x_{i}}

if Reμ_{i} = ∑ x_{i}λ_{i}. It follows that if λ = λ_{φ}(P, P^{∧}) then λ = Reμ_{1}. Thus P_{1} is determined by the condition that α^{∧} is a root of T ô in P ô ; 1 if and only if ⟨Reμ_{1}, α^{∧}⟩ ≥ 0.

P1^{∧} is therefore determined by φ alone. P_{1} is then any parabolic subgroup of G over R containing M whose class corresponds to that of P_{1}^{∧}. P and P^{∧} are then any pair with P ⊆ P_{1}, P^{∧} ⊆ P_{1}^{∧} whose classes correspond. Choose a Levi factor M_{1} of P_{1} over R that contains M. P ∩ M_{1} = P′ is a parabolic subgroup of M_{1}. If ρ ∈ Π_{φ}(P, P^{∧}) we may consider the irreducible constituents of I_{ρ}^{P′}, a representation of M_{1}(R). Let Π′_{φ} be the set formed by the infinitesimal equivalence classes of these constituents as ρ varies over Π_{φ}(P, P^{∧}). We have to observe that Π′_{φ} ⊆ Π(M_{1}) is independent of the choice of P and P^{∧}. Since Π′_{φ} is not affected if we simultaneously conjugate P′ and Π_{φ}(P, P^{∧}) with an element of M_{1}(R), we need only check that Π′_{φ} is independent of P^{∧}. But if we change P^{∧} then, by Lemma 2.5, we may change P and hence P′ so that Π_{φ}(P, P^{∧}) is not affected. Since I_{ρ}^{P′} and I_{ρ}^{\overline{P}′} are infinitesimally equivalent if P′ and \overline{P}′ are two parabolic subgroups of M_{1} with M as Levi factor, the set Π′_{φ} does depend only on φ.

We have normalized φ so that P1^{∧} contains Bô. Suppose φ′ is normalized in the same way and gives rise to the same P1^{∧}. We shall need to know that if Π′_{φ} and Π′_{φ′} have an infinitesimal equivalence class in common then φ and φ′ determine the same element of Φ(G). If Π′_{φ} and Π′_{φ′} have an element in common then, as we can see from our review of the results of [6], the images φ(W_{R}) and φ′(W_{R}) may be supported to lie in the same M^{∧}, no proper parabolic subgroup of which contains either of them. We may choose, tentatively, the same P and P^{∧} for both of them. Then there are a ρ in Π_{φ}(P, P^{∧}) and a ρ′ in Π_{φ′}(P, P^{∧}) and a g in the normalizer of M in M_{1}(R) such that ρ is equivalent to m → ρ′(gmg^{−1}). If ψ_{1} is the restriction of adg to M and ψ1^{∧} the associated automorphism of M^{∧} then, by condition (iv),

Π_{ψ_{1}^{∧} (φ)}(P, P^{∧}) = {m → ρ(gmg^{−1})|ρ ∈ Π_{φ}(P, P^{∧})}.

By Lemma 2.7, there is an h in the normalizer of M^{∧} in M ô ; 1 such that on M^{∧} the operator adh is equal to ψ1^{∧}. We may replace φ by adh ∘ φ. Then Π_{φ}(P, P^{∧}) and Π_{φ′}(P, P^{∧}) have an element in common; so φ and φ′ belong to the same class in Φ(M_{1}) and hence in Φ(G).

We are now able to introduce Π_{φ} in general. Π_{φ} consists of the classes J_{ρ}^{P_{1}}, ρ ∈ Π′_{φ}. By Lemma 3.14, together with the preceding discussion, these sets are disjoint. The other conditions on the sets Π_{φ} are built into their definition.

4. Exhaustion. It remains to prove the following proposition.

PROPOSITION 4.1. The sets Π_{φ}, φ ∈ Φ(G), exhaust Π(G).

We agree to call an infinitesimal equivalence class essentially tempered if it is a constituent of some I_{ρ} where ρ satisfies the conditions of Corollary 3.9 and is square-integrable modulo the center. To prove the proposition, we have only to prove the following Lemma.

LEMMA 4.2. If π is an irreducible quasi-simple representation of G(R), there is a parabolic subgroup P of G over R and an essentially tempered representation ρ of M(R), M = P/N, such that λ ∈ D^{+}(P) and such that π is infinitesimally equivalent to J_{ρ}.

λ has the same meaning as in Lemma 3.8. Notice that, by Lemma 3.14, ρ and P are uniquely determined by π. The lemma reduces the problem of classifying the classes of irreducible quasi-simple representations of G(R) to that of classifying the classes of essentially tempered representations of the various M(R).

Let π be given. The first, the easy, step is to find P. Let π act on V. If g ∈ G(R) and v^{∗} lies in the dual space of V define π^{∗}(g)v^{∗} by

⟨v, π^{∗}(g)v^{∗}⟩ = ⟨π(g^{−1})v, v^{∗}⟩.

If f ∈ C_{c}^{∞}(G) define π^{∗}(f)v^{∗} by

⟨v, π^{∗}(f)v^{∗}⟩ = ∫_{G(R)} f(g)⟨π(g^{−1})v, v^{∗}⟩dg.

It is clear that

∥π^{∗}(f)v^{∗}∥ ≤ ∥v^{∗}∥ ∫_{G(R)} |f(g)|∥π(g)∥ dg.

It is also clear that every K-finite vector v^{∗} is a finite linear combination

v^{∗} = ∑_{i} π^{∗}(f_{i})v_{i}^{∗}.

Let V^{∗} be the set of all v^{∗} for which

lim_{g→h} ∥π^{∗}(g)v^{∗} − π^{∗}(h)v^{∗}∥ = 0

for all h. Since V^{∗} contains all vectors of the form π^{∗}(f)v^{∗}, it contains all K-finite vectors. The representation π^{∗} on V^{∗} is continuous and the pairing (v, v^{∗}) → ⟨v, v^{∗}⟩ is G(R)-invariant.

Let 𝔛 be a finite collection of classes of irreducible representations of K. Let V (𝔛) and V^{∗}(𝔛) be the direct sum of the subspaces of V and V^{∗} transforming according to the representations in 𝔛. Consider the function Ψ from G(R) to the dual W (𝔛) of V (𝔛) ⊗ V^{∗}(𝔛) defined by

Ψ(g) : u ⊗ v → ⟨π(g)u, v⟩.

Choose a parabolic subgroup P_{0} of G minimal among those defined over R. We suppose that A ⊆ P_{0}(R). D(P_{0}) and the Lie algebra 𝔄 of A are in duality over R in such a way that

π_{λ}(expH) = e^{⟨λ,H⟩}.

It will be convenient to shuck some of our earlier notation, which is not always appropriate to our present purposes. Write D(P_{0}) = D_{0} + D, where D_{0} is orthogonal to 𝔄 ∩ 𝔤_{der} and D is orthogonal to 𝔄 ∩ 𝔤_{rad}. D has as basis the roots α_{1}, …, α_{r} of A simple with respect to P_{0}(R). The Killing form B(μ, ν) is nondegenerate on D and zero on D_{0}. Define β_{1}, …, β_{r} in D by

B(α_{i}, β_{j}) = δ_{ij}.

By Theorem 9.1.3.2 of [16] there is a countable subset L(π, 𝔛) in D(P_{0}) ⊗ C such that in the interior of A^{+} = A^{+}(P_{0}) an expansion

(4.1)  Ψ(a) = e^{−⟨δ,H⟩} ∑_{λ∈L(π,𝔛)} p_{λ}(H)e^{⟨λ,H⟩}

is valid. δ is one-half the sum of the roots of A positive with respect to P_{0} and taken with multiplicity. a = e^{H} and p_{λ} is a polynomial function of H with values in W (𝔛) that does not vanish identically.

If λ and μ belong to D(P_{0}) ⊗ C we write λ ≻ μ if

Reλ = Reμ + ∑_{i=1}^{r} x_{i}α_{i}

with x_{i} ≥ 0. Let E(π, 𝔛) be the set of λ maximal in L(π, 𝔛) with respect to this order. As in [16], E(π, 𝔛) is finite. There is a simple fact to be verified.

LEMMA 4.3. The set E(π, 𝔛) is the same for all 𝔛 for which W (𝔛) is not zero.

If 𝔛 and 𝔜 are two finite collections of classes of irreducible representations of K we may also introduce a function Ψ with values in the dual W (𝔛, 𝔜) of V (𝔛) ⊗ V^{∗}(𝔜) by

Ψ(g) : u ⊗ v → ⟨π(g)u, v⟩.

Thus Ψ also admits an expansion of the form (4.1). We introduce E(π, 𝔛, 𝔜) and show that it is the same for all 𝔛 and 𝔜 for which V (𝔛) ≠ 0 and V^{∗}(𝔜) ≠ 0.

It is clear that if 𝔛_{1} ⊆ 𝔛_{2} then every element of E(π, 𝔛_{1}, 𝔜) is dominated by an element of E(π, 𝔛_{2}, 𝔜). If V (𝔛) is different from 0 then every K-finite vector in V is a finite sum

∑ π(X_{i})u_{i}

with u_{i} ∈ V (𝔛), X_{i} ∈ 𝔜, the universal enveloping algebra of 𝔤. If

f(g) = ⟨π(g)u, v⟩

then

⟨π(g)π(X)u, v⟩ = Xf(g).

If u ∈ V (𝔛), v ∈ V^{∗}(𝔜) then ⟨π(g)π(X)u, v⟩ is a coordinate of the W (𝔛, 𝔜) valued function XΨ(g). It follows from Theorem 9.1.2.9 of [16] that every exponent in the expansion of ⟨π(g)π(X)u, v⟩ is dominated by an element of E(π, 𝔛, 𝔜). This if V (𝔛_{1}) ≠ 0 and V (𝔛_{2}) ≠ 0 then every element of E(π, 𝔛_{1}, 𝔜) is dominated by an element of E(π, 𝔛_{2}, 𝔜) and conversely. The two sets are therefore the same.

We define a double action w → τ_{1}(k_{1})wτ_{2}(k_{2}) of K on W (𝔛, 𝔜) by
τ_{1}(k_{1})wτ_{2}(k_{2}) : u ⊗ v → w(π(k_{1}^{−1})u ⊗ π^{∗}(k_{2})v).

We may interchange the roles of V and V^{∗} and of 𝔛 and 𝔜, replacing π by π^{∗}. If ω, the element of the Weyl group of A that takes positive roots to negative roots, is represented by k ∈ K, then

⟨u, π^{∗}(a)v⟩ = ⟨π(a^{−1})u, v⟩ = ⟨π(ω(a^{−1}))π(k)u, π^{∗}(k)v⟩.

Thus Ψ(a) is replaced by

τ_{1}(K^{−1})Ψ(ω(a^{−1}))τ_{2}(k)

and

E(π^{∗}, 𝔜, 𝔛) = {−ω(λ)|λ ∈ E(π, 𝔛, 𝔜)}.

It follows that E(π^{∗}, 𝔛, 𝔜) is also independent of 𝔜.

We all also need some simple geometric lemmas. We recall that B(α_{i}, α_{j}) ≤ 0 if i ≠ j and that B(β_{i}, β_{j}) ≥ 0 for all i and j. If F is a subset of {1, …, r} let D_{F} be the subspace of D spanned by {β_{i}|i ∈ F }. If i ∈ F let β_{i}^{F} = β_{i}; if i ∈ F let β_{i}^{F} be the orthogonal projection of β_{i} on the orthogonal complement of D_{F}. Define α_{i}^{F} by

B(α_{i}^{F} , β_{j}^{F}) = δ_{ij}.

If i ∉ F then α_{i}^{F} = α_{i}. If i ∈ F then

α_{i}^{F} = α_{i} + ∑_{k∉F} c_{ik}α_{k}.

If k is not in F then

0 = B(α_{i}^{F} , β_{k}^{F}) = (α_{i}, β_{k}^{F}) + c_{ik}.

{α_{k}|k ∉ F } is a basis for the orthogonal complement of D_{F} and B(α_{k}, α_{l}) ≤ 0 if k ≠ l. Since {β_{k}^{F} |k ∉ F } is the dual basis, B(β_{k}^{F} , β_{l}^{F} } ≥ 0. Therefore β_{l}^{F} is a linear combination of the α_{k} with nonnegative coefficients. Since B(α_{i}, α_{k}) ≤ 0, B(α_{i}, β_{k}^{F}) ≤ 0 for k ∉ F and c_{ik} ≥ 0. Thus if i and j belong to F and i ≠ j

B(α_{i}^{F} , α_{j}^{F}) = B(α_{i}^{F} , α_{j}) = B(α_{i}, α_{j}) + ∑_{k∉F} c_{ik}B(α_{k}, α_{j}) ≤ 0.

The inequality B(α_{i}^{F} , α_{j}^{F}) ≤ 0 is also valid if one of i and j does not lie in F.

For each F let ε_{F} be the characteristic function of

{λ ∈ D(P_{0})|B(α_{i}^{F} , λ) > 0, i ∈ F, B(β_{i}^{F} , λ) ≤ 0, i ∉ F }.

LEMMA 4.4. If λ ∈ D(P_{0}) then

∑_{F} ε_{F} (λ) = 1.

Suppose B(α_{i}, λ) > 0 for all i. Then B(α_{i}^{F} , λ) > 0 for all i and all F. Since the basis {β_{i}^{F} } is dual to {α_{i}^{F} } and B(α_{i}^{F} , α_{j}^{F}) ≤ 0 for i ≠ j, β_{i}^{F} is a linear combination of the α_{j}^{F} with nonnegative coefficients and B(β_{i}^{F} , λ) > 0 for all i and F. Thus ε_{F} (λ) = 0 unless F = {1, …, r} when ε_{F} (λ) = 1. Thus all we have to do is show that ε_{F} is a constant.

A hyperplane defined by an equation B(α_{i}^{F} , λ) = 0 or B(β_{i}^{F} , λ) = 0 for some i and F will be called special. If λ is any point in D(P_{0}) and if B(α_{i}, μ) > 0 for all i, then for any sufficiently small positive real number a

ε_{F} (λ) = ε_{F} (λ − aμ)

for all F. Moreover λ − aμ lies in no special hyperplane. To show that ε_{F} is a constant we have to show that it is constant on the complement of the special hyperplanes. For this we have only to verify that it is constant in a neighborhood of a point λ_{0} lying in exactly one special hyperplane.

For this we may disregard all those F which lie neither in

S_{1} = {F |B(α_{i}^{F} , λ_{0}) = 0 for some i ∈ F }

nor in

s_{2} = {F |B(β_{i}^{F} , λ_{0}) = 0 for some i ∉ F }.

The sets S_{1} and S_{2} are disjoint. F = {1, …, r} does not belong to S_{2}. We can introduce a bijection between S_{1} and S_{2}. If F_{1} ∈ S_{1} and α_{i}^{F} with i ∈ F_{1} is orthogonal to λ_{0} set F_{2} = F_{1} − {i}. α_{i}^{F_{1}} and β_{i}^{F_{2}} both lie in the space spanned by {β_{j}|j ∈ F_{1}} and are both orthogonal to {β_{j}|j ∈ F_{2}}. Thus they are multiples of each other and F_{2} ∈ S_{2}. It is clear that F_{1} → F_{2} is a bijection. Since

1 = B(α_{i}^{F_{1}}, β_{i}) = B(α_{i}^{F_{1}}, β_{i}^{F_{2}}),

α_{i}^{F_{1}} is a positive multiple of β_{i}^{F_{2}}. We have relations

α_{j}^{F_{2}} = a_{j}^{F_{1}} + c_{j}α_{i}^{F_{1}}, j ∈ F_{2},
β_{j}^{F_{2}} = β_{j}^{F_{1}} + d_{j}β_{i}^{F_{2}}, j ∉ F_{1}.

Near λ_{0}

signB(β_{j}^{F_{2}}, λ) = signB(β_{j}^{F_{1}}, λ), j ∉ F_{1},
signB(α_{i}^{F_{2}}, λ) = signB(α_{j}^{F_{1}}, λ), j ∈ F_{2}.

Moreover, either B(α_{i}^{F_{1}}, λ) > 0 or B(β_{i}^{F_{2}}, λ) ≤ 0 but not both. Thus ε_{F_{1}} + ε_{F_{2}} is constant near λ_{0}. The lemma follows.

If λ ∈ D(P_{0}) let F = F (λ) be the unique subset of {l, …, r} such that

B(α_{i}^{F} , λ) > 0, i ∈ F,
b(β_{i}^{F} , λ) ≤ 0, i ∉ F.

Let λ^{0} be the projection of λ on the sum of D_{0} and D_{F}. Then B(α_{i}, λ^{0}) ≥ 0 for all i and B(α_{i}, λ^{0}) > 0 if i ∈ F. This is clear because B(α_{i}, λ^{0}) = 0 if i ∉ F and B(α_{i}, λ^{0}) = B(α_{i}^{F} , λ^{0}) = B(α_{i}^{F} , λ) if i ∈ F. Let λ = λ^{0} + λ^{1}. Then

λ^{1} = ∑_{i∉F} b_{i}α_{i}.

Notice that

b_{i} = B(β_{i}, λ^{1}) = B(β_{i}^{F} , λ^{1}) = B(β_{i}^{F} , λ) ≤ 0.

LEMMA 4.5. Suppose λ and μ lie in D(P_{0}) and

λ^{0} + ∑_{i=1}^{r} c_{i}α_{i} = μ^{0} + ν + ∑_{j∉F (μ)} d_{j}α_{j}.

Suppose c_{i} ≤ 0, ν ∈ D, B(α_{i}, ν) = 0 if i ∉ F (μ), and B(β_{i}, ν) ≥ 0 if i ∈ F (μ). Then λ^{0} ≻ μ^{0}.

Certainly λ^{0} − μ^{0} ∈ D. If i ∈ F (μ) then
(4.2)  B(β_{i}, λ^{0} − μ^{0}) = −c_{i} + B(β_{i}, ν) ≥ 0.

If i ∉ F (μ) then

B(α_{i}, λ^{0}) = B(α_{i}, λ^{0}) ≥ 0.

If i ∉ F (μ)

β_{i}^{F} = ∑_{j∉F (μ)} e_{j}α_{j}

with e_{j} ≥ 0; so

B(β_{i}^{F} , λ^{0} − μ^{0}) ≥ 0.

Moreover

β_{i}^{F} = β_{i} − ∑_{j∈F (μ)} a_{j}β_{j}

and

a_{j} = B(β_{i}, β_{j})|B(β_{j}, β_{j}) ≥ 0.

For (4.2) we conclude that

B(β_{i}, λ^{0} − μ^{0}) ≥ B(β_{i}^{F} , λ^{0} − μ^{0}) ≥ 0.

The lemma follows.

COROLLARY 4.6. iF λ ≻ μ then λ^{0} ≻ μ^{0}.

If λ ≻ μ then

λ + ∑_{i=1}^{r} c_{i}α_{i} = μ

with c_{i} ≤ 0. Since

λ = λ^{0} + ∑_{i∉F (λ)} b_{i}α_{i}

with b_{1} ≤ 0 and

μ = μ^{0} + ∑_{j∉F (μ)} d_{j}α_{j}

the corollary follows.

Since the set E(π, 𝔛) is the same for all 𝔛 with W (𝔛) ≠ 0 we may denote it by E(π). Consider

L^{0}(π, 𝔛) = {λ^{0}|λ = Reλ′, λ′ ∈ L(π, 𝔛)}

and

E^{0}(π) = {λ^{0}|λ = Reλ′, λ′ ∈ E(π)}.

Suppose μ^{0} lies in L^{0}(π, 𝔛). There is a λ′ ∈ E(π) such that λ′ ≻ μ′; then λ = Reλ′ ≻ μ = Reμ′ and λ^{0} ≻ μ^{0}. Thus L^{0}(π, 𝔛) has a maximal element λ^{0} and λ^{0} ∈ E^{0}(π). We fix such a λ^{0} once and for all. Since λ^{0} lies in the closure of D^{+}(P_{0}) there is a unique P containing P_{0} such that λ^{0} ∈ D^{+}(P). This will turn out to be the P which appears in Lemma 4.2.

To obtain the representation ρ we have to apply some results that appear in an unpublished manuscript of Harish-Chandra [5] but, to the best of my knowledge, nowhere else.

D(P_{0}) is the sum of D_{1} = D(P) and its orthogonal complement D_{2}. A is a product A_{1}A_{2}, where A_{1} = A(P) = {e^{H}|H⊥D_{2}} and A_{2} = {e^{H}|H⊥D_{1}}. Let L_{1}(π, 𝔛) be the projection of L(π, 𝔛) on D_{1} ⊗ C. The first result we need from [5] is that Ψ(a) = Ψ(a_{1}, a_{2}) = Ψ(e^{H_{1}}, a_{2}) admits an expansion

(4.3)  e^{−⟨δ,H_{1}⟩} ∑_{λ_{1}∈L_{1}(π,𝔛)} ϕ_{λ_{1}}(H_{1}, a_{2})e^{⟨λ_{1},H_{1}⟩}

valid for a_{1} in the interior of A_{1}^{+} = A^{+}(P). ϕ_{λ}(H_{1}, a_{2}) is a polynomial function of H_{1} whose coefficients are analytic functions of a_{2}. The degrees of thee polynomials are bounded. If a = e^{H} ∈ A^{+} = A^{+}(P_{0}) then

(4.4)  ϕ_{λ}(H_{1}, a_{2})e^{⟨λ_{1}−δ,H_{1}⟩} = ∑ p_{λ}(H)e^{⟨λ−δ,H⟩}

where the sum is taken over all λ ∈ L(π, 𝔛) whose projection on D_{1} ⊗ C is λ_{1}.

To exploit this expansion we have to generalize some considerations to be found in §9.1.2 of [16]. The generalization being quite formal, we shall be as sparing as possible with proofs.

Choose a Levi factor M of P over R such that M(R) is selfadjoint. Let 𝔭, 𝔪, 𝔫, 𝔨 in 𝔤 be the complexifications of the Lie algebras of P (R), M(R), N(R), and K and let 𝔓, 𝔐, 𝔑, and 𝔎 be their universal enveloping algebras. Let 𝔮 be the orthogonal complement of 𝔪 in 𝔨 ∩ 𝔤_{der}. As on p. 269 of [16], but with a different result, we define 𝔔 to be the image of the symmetric algebra of 𝔮 in 𝔄.

Note that

dim 𝔮 = dim 𝔤 − dim 𝔭 = dim 𝔫

and that

dim 𝔤 = 2 dim 𝔮 + dim 𝔪.

Let K_{1} = K ∩ M(R). It is a maximal compact subgroup of M(R). Let U be a compact subset of M(R) with U = K_{1}U = UK_{1}. As m varies over U the eigenvalues of adm in the orthogonal complement of 𝔪 in 𝔤_{der} lie in compact subset of C^{×}, say

\{z|\frac{1}{R} ≤ |z| ≤ R\}.

Let A_{1}^{+}(R) be the set of all a in A_{1} such that α(a) > R for every root of A_{1} in 𝔫. If m = m_{1}a, m_{1} ∈ U, a ∈ A_{1}^{+}(R) the centralizer of m in 𝔤 lies in m. Moreover

(4.5)  𝔤 = ad m(𝔮) ⊕ 𝔪 ⊕ 𝔮.

To see this one has only to verify that

ad m(𝔮) ∩ (𝔪 + 𝔮) = 0.

Since 𝔪 and 𝔮 are invariant under K_{1} and M(R) = K_{1}AK_{1} we may suppose that m_{1}, and hence m, lies in A. Suppose X lies in the above intersection. Let θ be the automorphism of G(C) such that θ(g^{−1}) is the conjugate transpose of g with respect to the hermitian form introduced earlier. θ is a Cartan involution. Let H lie in the Lie algebra of A_{1} and set

X_{H} = (adH)^{2}X.

Then X_{H} ∈ 𝔨 and ad m(X_{H}) ∈ 𝔨. Consequently

ad m(X_{H}) = θ(ad m(X_{H})) = ad m^{−1}(X_{H})

and

ad m^{2}(X_{H}) = X_{H}.

Since ad m has only positive eigenvalues and since its centralizer in 𝔤 is 𝔪, this equation implies that X_{H} ∈ 𝔪. Thus

(ad H)^{3}X = ad H(X_{H}) = 0.

However, ad H is semisimple; so X_{H} = 0. Since H was arbitrary in the Lie algebra of A_{1}, X lies in 𝔪. Since both 𝔪 ∩ 𝔮 and 𝔪 ∩ ad m(𝔮) must be zero, X is zero.

The relation (4.5) yields an isomorphism

𝔄 ≃ ad m(𝔔) ⊗ 𝔐 ⊗ 𝔔 ≃ 𝔔 ⊗ 𝔐 ⊗ 𝔔.

If X ∈ 𝔄 we let X_{m} be the corresponding element on the right. The function Ψ restricted to M(R) yields a function on M(R) with values in W (𝔛). If X ∈ 𝔐 we denote the result of applying X to this function at the point m by Ψ(m, X). The actions τ_{1} and τ_{2} of K on W (𝔛) yield actions of 𝔨. Let X → X^{∼} be the involution of 𝔨 defined by X^{∼} = −X, X ∈ 𝔨. If X ∈ 𝔄 and

X_{m} = ∑ X_{i} ⊗ Y_{i} ⊗ Z_{i}

then

XΨ(m) = ∑ τ_{1}(Z_{i}^{∼})Ψ(m, Y_{i})τ_{2}(Z_{i}^{∼}).
Let \overline{P} = θ(P). Then \overline{P} is defined over R and \overline{P} ∩ P = M. Moreover
𝔤 = \overline{𝔫} + 𝔪 + 𝔮

and

𝔄 = \overline{𝔑}𝔐𝔔.

If X = ∑ Y_{i}Z_{i}, Y_{i} ∈ 𝔐, Z_{i} ∈ 𝔔 then

X_{m} = ∑ 1 ⊗ Y_{i} ⊗ Z_{i}.

Suppose \overline{X} ∈ \overline{𝔫}. Let \overline{X} = θ(X), X ∈ 𝔫. Then

Y = X + \overline{X}

lies in 𝔮. Let X′ = adm(X), \overline{X}′ = θ(X′) and

Y′ = X′ + \overline{X}′ = X′ + θ(adm(X)).

Since

ad m(Y) = X′ + ad m(\overline{X}) = X′ + θ(adθ(m)(X))

we have

θ({ad m − ad θ(m)}X) = Y′ − ad mY.

We are still assuming that m = m_{1}a, m_{1} ∈ U, a ∈ A_{1}^{+}(R); the restriction of ad m − ad θ(m) to 𝔫 is therefore invertible. Let 𝔉 be the ring of functions generated by the matrix coefficients of its inverse. 𝔉 does not contain 1. Replacing X by {ad m − ad θ(m)}^{−1}X, we see that

\overline{X} = ∑ f_{i}(m)ad m(X_{i}) + ∑ g_{i}(m)Z_{i}

with f_{i}, g_{i} in 𝔉 and X_{i}, Z_{i} in 𝔔. Then

\overline{X}_{m} = ∑ f_{i}(m)X_{i} ⊗ 1 ⊗ 1 + ∑ g_{i}(m)1 ⊗ 1 ⊗ Z_{i}.

One proves more generally by induction on the degree that

(4.6)  X_{m} = X_{0} + ∑ f_{i}(m)X_{i}

where f_{i} ∈ 𝔉, X_{i} ∈ 𝔔 ⊗ 𝔐 ⊗ 𝔔 and where X_{0} ∈ 𝔐 ⊗ 𝔔 ≃ 𝔐𝔔 is uniquely defined by the conditions that X − X_{0} ∈ \overline{𝔫}𝔄.

Notice that as a function of a ∈ A_{1}^{+} an element of 𝔉 is a linear combination of products of the functions {α(a) − α^{−1}(a)}^{−1}, α a root of A_{1} in 𝔫 with coefficients that are analytic functions of m_{1}. Moreover {α(a) − α^{−1}(a)}^{−1} admits an expansion.

(4.7)  ∑_{n=0}^{∞} e^{−(2n+1)⟨α,H⟩}

for a = e^{H} in A_{1}^{+}(R).

If X ∈ D, the centralizer of K in 𝔄, and if M(X) is the linear transformation of W (𝔛) adjoint to the operator

u ⊗ v → π(X)u ⊗ v

on V (𝔛) ⊗ V^{∗}(𝔛) then

XΨ = M(X)Ψ.

λ^{0} ∈ D_{1} was fixed some time ago. There is at least one λ_{1}^{0} ∈ L_{1}(π, 𝔛) with Reλ_{1}^{0} = λ^{0}. Fix such a λ_{1}^{0}. If m ∈ M(R) we write m = k_{1}ak_{2} with k_{1}, k_{2} in K_{1} and a in A. We write a = a_{1}a_{2}, a_{1} = e^{H_{1}}, and set

Φ(m) = e^{⟨λ_{1}^{0}−δ,H⟩}τ_{1}(k_{2}^{−1})ϕ_{λ_{1}^{0}}(H_{1}, a_{2})τ_{2}(k_{1}^{−1}).

Because of the uniqueness of the expansion (4.3), Φ is well defined. The elements of 𝔎 ⊗ 𝔐 ⊗ 𝔎 act on Φ. X ⊗ Y ⊗ Z sends Φ to Φ′ with

Φ′(m) = τ_{1}(Z^{∼})Φ(m, Y)τ_{2}(X^{∼}).

Let X ∈ 𝔇 and let X_{0} be defined as in (4.6); then X_{0} ∈ 𝔎 ⊗ 𝔐 ⊗ 𝔎 and

(4.8)  X_{0}Φ = M(X)Φ.

To see this we start from the equation XΨ(m) = M(X)Ψ(m). If we set m_{2} = k_{1}a_{2}k_{2} and

ϕ_{λ_{1}}(H_{1}, m_{2}) = τ_{1}(k_{2}^{−1})ϕ_{λ_{1}}(H_{1}, a_{2})τ_{2}(k_{1}^{−1})

the function M(X)Ψ(m) has an expansion

∑_{λ_{1}∈L_{1}(π,X)} M(X)ϕ_{λ_{1}}(H, m_{2})e^{⟨λ_{1}−δ,H⟩}.

The function XΨ(m) is equal to

X_{0}Ψ(m) + ∑ f_{i}(m)X_{i}Ψ(m).

X_{0} and the X_{i} are acting as elements of 𝔎 ⊗ 𝔐 ⊗ 𝔎. Because of (4.7) each f_{i}(m) has an expansion

∑_{μ_{1}} ε_{μ_{1}}(m_{2})e^{−⟨μ_{1},H_{1}⟩}

valid for m_{2} ∈ U, a_{1} ∈ A_{1}^{+}(R), where U is a compact set in M(R) and R = R(U) is chosen as before. μ_{1} runs over the projections on D_{1} of sums of positive roots of A in 𝔫. The sums are not empty and μ_{1} is never zero. We may, for convergence offers no difficulty (cf. 16), apply X_{i} to Ψ term by term, expand the product f_{i}(m)X_{i}Ψ(m) formally, add the results, and then compare coefficients of the exponentials e^{⟨λ_{1}−δ,H_{1}⟩} on both sides of the equation.

We are interested in the terms corresponding to λ_{1}^{0}. If we incorporate the exponential, the term on the right is M(X)Φ(m). At first sight the term on the left seems more complicated. Suppose, however, that μ_{1} is the projection on D_{1} of a sum of positive roots of A in 𝔫, ν_{1} lies in L_{1}(π, 𝔛), and ν_{1} − μ_{1} = λ_{1}^{0}. Let λ_{1}^{0} be the projection of λ′ in L(π, 𝔛) and let λ = Reλ′; let ν_{1} be the projection of ν′ and let ν = Reν′. Then

Reλ_{1}^{0} = λ^{0}

and, if as before we define ν^{0} to be the projection of ν on the sum of D_{0} and D_{F (ν)}, then

Reν_{1} = ν + ∑_{j∉F (λ)} c_{j}α_{j} = ν^{0} + ∑_{i=1}^{r} b_{i}α_{i} + ∑_{j∉F (λ)} c_{j}α_{j}

with b_{i} ≤ 0. Also

μ_{1} = ∑_{i=1}^{r} d_{i}α_{i} + ∑_{j∉F (λ)} e_{j}α_{j}

with d_{i} ≥ 0. Moreover, at least one d_{i}, with i ∈ F (λ), is positive. We have

ν^{0} + ∑_{i=1}^{r}(b_{i} − d_{i})α_{i} = λ^{0} + ∑_{j∉F (λ)}(e_{j} − c_{j})α_{j}.

It follows from Lemma 4.5 that ν^{0} ≻ λ^{0}. By the very choice of λ^{0}, ν^{0} is therefore equal to λ^{0}. However if i ∈ F (λ) then

B(β_{i}, ν^{0} − λ^{0}) = d_{i} − b_{i}.

Since this is positive for at least one i, ν^{0} ≠ λ^{0}. This is a contradiction. The term on the left in which we are interested is therefore X_{0}Φ(m). The relation (4.8) follows.

𝔇 contains ℨ. As a linear space 𝔄 is a sum

𝔐 + \overline{𝔫}𝔐 + 𝔐𝔫 + \overline{𝔫}𝔐𝔫

and

ℨ ⊆ 𝔐 + \overline{𝔫}𝔐𝔫.

Thus if X ∈ ℨ = ℨ_{G} then X_{0} belongs to 𝔐 and in fact to ℨ_{M}. The map X → X_{0} is an injection of ℨ_{G} into ℨ_{M} and turns ℨ_{M} into a finite ℨ-module. Notice also that M(X) is a scalar m(X)I if X ∈ ℨ_{G}.

According to (4.4) the restriction of Φ to A has an asymptotic expansion ∑ p_{λ}(H)e^{⟨λ−δ,H⟩} where λ runs over those elements of L(π, 𝔛) whose projection on D_{1} ⊗ C is λ_{1}^{0}. Suppose ν′ is one of the indices for this sum. Let ν = Re ν′ and define ν^{0} as before. We can again apply Lemma 4.5 to see that ν^{0} = λ^{0}. Thus if F = {i|B(α_{i}, λ^{0}) > 0} then F = F (ν) and

(4.9)  Re{B(β_{F}^{i} , ν)} ≤ 0, i ∉ F.

In spite of the fact that Φ is not an eigenfunction of ℨ_{M} but only of the image of ℨ_{G} in ℨ_{M} the considerations of §9.1.3 of [16], and hence those of its appendix as well as those of [5], may be applied to it. We do not want to apply them to obtain an asymptotic expansion, which we already have; we want to apply a further result (Theorem 4) of [5] that in conjunction with (4.) and Lemma 3.7 easily implies the existence of a constant c and an integer d such that

(4.10)  π_{δ_{1}}(m)∥Φ(m)∥ ≤ c(1 + l(m))^{d}π_{λ^{0}}(m)Ξ_{M}(m)

for all m. δ_{1} is the projection of δ on D_{1}.

We had fixed 𝔛 but we may let it grow without changing λ^{0}. Thus

Φ(m)(u ⊗ v) = Φ(m; u, v)

is defined for all K-finite μ ∈ V, v ∈ V^{∗}.

LEMMA 4.7. Suppose v in V^{∗} is K-finite. If the function Φ(m; π(k)u, v) vanishes identically in m and k for some nonzero K-finite u in V then it vanishes identically for all such u.

The function ϕ(m) = ⟨π(m)u, v⟩, m = a_{1}m_{2}, a_{1} = e^{H_{1}} ∈ A_{1}, m_{2} = k_{1}a_{2}k_{2}, a_{2} ∈ A_{2}, k_{1}, k_{2} ∈ K_{1}, admits an asymptotic expansion ∑ a_{λ_{1}}(m; u, v)e^{⟨λ_{1}−δ,H_{1}⟩} with

a_{λ_{1}^{0}}(m; u, v)e^{⟨λ_{1}^{0}−δ,H_{1}} = Φ(m, u, v).

Suppose X ∈ 𝔄 and write

X_{m} = X_{0} + ∑ f_{i}(m)X_{i}.

For this we have to constrain m_{2} to vary in some compact set U and a_{1} to vary in A_{1}^{+}(R), R = R(U). Then

⟨π(m)π(X)u, v⟩ = Xϕ(m) = X_{0}ϕ(m) + ∑_{i} f_{i}(m)X_{i}ϕ(m).

The symbol Xϕ(m) denotes the value of X applied to the function ϕ(g) = ⟨π(g)u, v⟩ at the point m. X_{0} and X_{i} are applied as elements of 𝔎 ⊗ 𝔐 ⊗ 𝔎.

The considerations used to prove the equality (4.8) show that if u′ = π(X)u then a_{λ_{1}^{0}}(m; u′, v) is the coefficient of e^{⟨λ_{1}^{0}−δ,H_{1}⟩} in the expansion of X_{0}ϕ(m).

X_{0} = ∑ 1 ⊗ Y_{j} ⊗ Z_{j}

with Y_{j} ∈ 𝔐, Z_{j} ∈ 𝔎. Applying Z_{j} we replace the coefficient a_{λ_{1}^{0}}(m, u, v) by a_{λ_{1}^{0}}(m, π(Z_{j})u, v). If Φ(m, π(k)u, v) = 0 for all m and k, this is zero. If the coefficient is zero before Y_{j} is applied it is zero after. Since every K-finite vector in V is of the form π(X)u, X ∈ 𝔄, the lemma follows.

There is certainly at least one K-finite v in V^{∗}, which we fix once and for all, such that Φ(m; u, v) is not zero for all K-finite u.

Let 𝔗 be the Banach space of continuous functions θ on M(R) for which

∥θ∥ = sup\frac{|θ(m)|}{(1 + l(m))^{d−1}π_{λ^{0}}(m)Ξ_{M}(m)} < ∞.

If m ∈ M(R) let r(m)θ be the function whose value at m_{1} is

θ(m_{1}m)

Let 𝔚 be the space of all θ in 𝔗 for which

lim_{m→m_{0}} ∥r(m)θ − r(m_{0})θ∥ = 0

for all m_{0}. If u ∈ V is K-finite then

θ_{u} : m → π_{δ_{1}}(m)Φ(m; u, v)

lies in 𝔚 because of (4.10). Let 𝔙 be the closed subspace of 𝔚 generated by the functions r(m)θ_{u}.

LEMMA 4.8 The representation r of M(R) on 𝔙 admits a finite composition series.

Let 𝔙_{0} be the space of functions in 𝔙 of the form

θ = r(f)θ′ = ∫_{M(R)} f(m)r(m)θ′dm

with f ∈ C_{c}^{∞}(m(R)). If X ∈ ℨ_{G} ↪ ℨ_{M} and θ ∈ 𝔙_{0} then

(4.11)  Xθ = m′(X)θ

if m′(X) = m(X′), where X′ is the element of ℨ_{M} defined by

X(π_{δ_{1}}ϕ)(m) = π_{δ_{1}}(m)X′ϕ(m).
If K_{1} = K ∩ M(R), if φ_{1} and φ_{2} are two continuous functions on K_{1}, and if θ ∈ 𝔙_{0} let
θ(m; φ_{1}, φ_{2}) = ∫_{K_{1}} ∫_{K_{1}} φ_{1}(k_{1})θ(k_{1}mk_{2})φ_{2}(k_{2})dk_{1}dk_{2}.

If φ′_{1}(k) = φ_{1}(kk_{1}^{−1}) and φ′_{2}(k) = φ_{2}(k_{2}^{−1}k) then

θ(k_{1}mk_{2}; φ_{1}, φ_{2}) = θ(m, φ′_{1}, φ′_{2}).

If θ = r(m_{1})θ_{u} then

θ(m, φ_{1}, φ_{2}) = π_{δ_{1}}(m) ∫ ∫ φ_{1}(k_{1})Φ(k_{1}mm_{1}k_{2}; u, v)φ_{2}(k_{2})dk_{1}dk_{2}
= π_{δ_{1}}(m) ∫ ∫ φ_{1}(k_{1})Φ(mm_{1}, π(k_{2})u, π^{∗}(k_{1}^{−1})v)φ_{2}(k_{2})dk_{1}dk_{2}
= π_{δ_{1}}(m)Φ(mm_{1}, u′, v′)

with

u′ = ∫ φ_{2}(k_{2})π(k_{2})udk_{2}, v′ = ∫ φ_{1}(k_{1})π^{∗}(k_{1}^{−1})vdk_{1}.

In particular, if v′ = 0 then θ(m, φ_{1}, φ_{2}) = 0 for θ = r(m_{1})θ_{u} and hence, by continuity, for any θ in 𝔙_{0}. There is a closed subspace of finite codimension in the space of continuous functions on K_{1}, invariant under left and right translations, such that v′ = 0 whenever φ_{1} lies in this subspace. Factoring out the subspace, we may regard φ_{1} as varying over a finite-dimensional space. Let 𝔛_{1} be a finite set of classes of irreducible representations of K_{1}. For other, more obvious, reasons, if θ is constrained to lie in the subspace 𝔙(𝔛_{1}) of 𝔙_{0} spanned by vectors transforming according to one of the representations in 𝔛_{1}, then φ_{2} may be regarded as varying over a finite-dimensional space. Using (4.11) and a simple variant of Proposition 9.1.3.1 of [16], we conclude that the space of functions m → θ(m; φ_{1}, φ_{2}), where θ ∈ 𝔙(𝔛_{1}) and φ_{1} and φ_{2} are continuous functions on K_{1}, is finite-dimensional. Since φ_{1} and φ_{2} may be allowed to approach the delta-function, we conclude that 𝔙(𝔛_{1}) itself lies in this space and is finite-dimensional. Since 𝔙_{0} is dense in 𝔙 every irreducible representation of K_{1} occurs with finite multiplicity in 𝔙.

To complete the proof we need a well-known fact, which we state as a lemma.

LEMMA 4.9 Let X → m′(X) be homomorphism of ℨ_{G} ↪ ℨ_{M} into C. There are only a finite number of infinitesimal equivalence classes of quasi-simple irreducible representations τ of M(R) such that

τ(X) = m(X)I

for X ∈ ℨ_{G}.

Since there are only a finite number of ways of extending m to a homomorphism of ℨ_{M} into C, it is enough to prove the lemma for G = M; that is, we may assume that m′ is already given on ℨ_{M} and that

τ(X) = m′(X)I

for all X ∈ ℨ_{M}.

Let τ act on W. We saw in Lemma 3.5 that the restriction of τ to the connected component M^{0}(R) is the direct sum of finitely many irreducible representations. Let τ^{0}, acting on W^{0} ⊆ W, be one of them. Because of Theorem 4.5.8.9 of [16], there are only finitely many possibilities for the class of τ^{0}.

Suppose W′ is the space of all functions φ on M(R) with values in W^{0} satisfying

φ(m_{0}m) = τ^{0}(m_{0})φ(m), m_{0} ∈ M^{0}(R).

M(R) acts on W′ by right translations. There is an M(R)-invariant map from W′ to W given by

φ → ∑_{M^{0}(R)\M(R)} τ(g^{−1})φ(g).

We shall verify that W′ admits a finite composition series

0 = W_{0} ⊈ W_{1} ⊈ W_{2} ⊈ … ⊈ W_{n} = W′.

Then τ must be equivalent to the representation of M(R) on one of the quotients W_{i−1}\W_{i}. From this lemma follows.

To show the existence of a finite composition series all we have to do is show that if

0 = W_{0} ⊈ W_{1} ⊈ … ⊈ W_{n} = W′

is any chain of M(R)-invariant subspaces then n ≤ [M(R) : M^{0}(R)]. We could instead work with spaces of K-finite vectors invariant under the pair K, 𝔐. If K^{0} = K ∩ M^{0}(R) then W′ admits a composition series of length [M(R) : M^{0}(R)] with respect to the pair K^{0}, 𝔐. Any chain invariant with respect to this pair, and, a fortiori, any chain invariant with respect to K, 𝔐, has therefore length at most [M(R) : M^{0}(R)].

We return to the proof of Lemma 4.8. Let τ_{1}, …, τ_{s} be the classes corresponding to the given homomorphism m′. Choose for each i an irreducible representation σ_{i} occurring in the restriction of τ_{i} to K_{1}. Set 𝔛_{1} = {σ_{1}, …, σ_{s}}.

Suppose 𝔙′′ ⊈ 𝔙′ are closed M(R) -invariant subspaces of 𝔙. Let σ be a representation of K occurring in 𝔚^{0} = 𝔇′′\𝔇′. Let 𝔚(σ) be the space of all vectors in 𝔚^{0} transforming according to σ. 𝔚(σ) is finite-dimensional. Among the nonzero subspaces of 𝔚(σ) obtained by intersecting it with a closed M(R) -invariant subspace of 𝔚^{0}, there is a minimal one 𝔚′(σ). Let 𝔚′ be the intersection of all closed invariant subspaces of 𝔚^{0} that contain 𝔚′(σ). Let 𝔚′′ be the closure of the sum of all closed invariant subspaces of 𝔚′ that do not contain 𝔚′(σ). Then 𝔚′′ ⊈ 𝔚′ and the representation of M(R) on 𝔚′′\𝔚′ is irreducible. Since it must be one of τ_{1}, …, τ_{s}, it contains one of σ_{1}, …, σ_{s}.

Suppose we have a chain of closed M(R)-invariant subspaces

0 ⊈ 𝔙_{1} ⊈ … ⊈ 𝔙_{n} = 𝔙.

Since one of σ_{1}, …, σ_{s} is contained in the representation of K_{1} on the quotient of the successive subspaces, n ≤ dim 𝔙(𝔛_{1}). On the other hand, if these quotients are not irreducible the chain can be further refined. The lemma follows.

As before let \overline{P} = θ(P). Let 𝔄 be the space of continuous functions φ on G(R) with values in 𝔙 which satisfy the following two conditions:

(i) If \overline{n} ∈ \overline{N}(R) then φ(\overline{n}g) = φ(g).

(ii) If m ∈ M(R) then φ(mg) = π_{δ_{1}}^{−1}(m)r(m)φ(g). The representation of G(R) on 𝔘 by right translations is the induced representation I\overline{P} ; r. It is easily seen that every representation of K occurs with finite multiplicity in I\overline{P} ; r and that

I\overline{P} ; r (X) = m(X)I, I ∈ ℨ_{G}.

Thus I\overline{P} ; r admits a finite composition series. We now show that π is infinitesimally equivalent to a subrepresentation of I_{r}\overline{P}. For this we have only to define an injection of the K-finite vectors in V into 𝔄 which commutes with the action of K and 𝔄.

Recall that the vector v was fixed. Suppose u is K-finite. If k_{1} ∈ K_{1} then

Φ(mk_{1}^{−1}, π(k_{1}k)u, v) = Φ(m, π(k)u, v).

We define φ_{u} in 𝔘 by

(4.12)  φ_{u}(\overline{n}mk) : m_{1} → π_{δ_{1}}(m_{1})Φ(m_{1}m, π(k)u, v).

The map u → φ_{u} is by our choice of v, an injection; it clearly commutes with the action of K. To verify that it commutes with the action of 𝔄 we have only to check that

φ_{π(X)u}(1) = (I\overline{P} ; r (X)φ_{u})(1).

Set φ_{u} = φ and φ_{π(X)u} = φ′. Then φ(m) = Φ(m; u, v) and φ′(m) = Φ(m, π(X)u, v). Recall that if X_{0} is defined as in (4.6) and equals

∑ 1 ⊗ Y_{i} ⊗ Z_{i}

then

φ′(m) = ∑ Y_{i}φ_{i}(m).

On the right Y_{i} is applied to a function of m and

φ_{i}(m) = Φ(m, π(Z_{i})u, v).

X_{0} was so chosen that

X − ∑ Y_{i}Z_{i} ∈ \overline{n}𝔄

It is clear that if Y ∈ \overline{𝔫} and ψ is K-finite in 𝔘 then

I\overline{P} ; r (Y)ψ(1) = 0.

Thus

I\overline{P} ; r (X)φ(1) = ∑ I\overline{P} ; r (Y_{i})I\overline{P} ; r (Z_{i})φ(1) = ∑ I\overline{P} ; r (Y_{i})φ_{π(Z_{i})u}(1).

A close examination of the definition (4.12) shows that I\overline{P} ; r (Y_{i})φ_{π(Z_{i})u}(1) is the function m → Y_{i}φ_{i}(m).

There must be an irreducible constituent ρ of the representation r on 𝔙 such that π is infinitesimally equivalent to a subrepresentation of I\overline{P} ; ρ. This ρ is the representation figuring in Lemma 4.2, which we are still in the process of proving. We must show that ρ is essentially tempered. Accepting this for the moment, we show that π is infinitesimally equivalent to the representation J_{ρ}^{P}.

An easy computation (for a special case, see Chapter 5 of ([16]) shows that I_{ρ}^{P} and I\overline{P} ; ρ have the same character and therefore the same irreducible constituents.

Let ρ act on W. J_{ρ}^{P} was introduced as the representation on the quotient I_{0}(W)/I_{1}(W). All we have to do is verify that π cannot be a constituent of the restriction of I_{ρ}^{P} to I_{1}(W).

The λ = λ(ρ) that figures in Lemma 3.8 is λ^{0}. If π is a constituent of the restriction of I_{ρ}^{P} to I_{1}(W) then, by Lemma 3.12,

(4.13)  ⟨π(am)u, v⟩ = o(δ_{P}^{−1}(a)π_{λ_{0}}(a))

if m is fixed in M(R) and a → ∞ in A^{+}(P). However, Theorem 3 of [5] assures us that the expansion (4.3) converges decently for fixed a_{2} (cf. [16]), Appendix 3). We conclude from (4.13) and Lemma A.3.2.3 of [16] that the terms of (4.3) with Reλ_{1} = λ^{0}. This certainly contradicts the choice of λ^{0}.

We apply Lemma A.3.2.3 in the following manner. Choose λ_{1}^{0} ∈ L_{1}(π, 𝔛) with Reλ_{1}^{0} = λ^{0}. Let a_{2} be fixed. If a_{1} = e^{H_{1}} lies in A_{1} then δ_{P} (a_{1}) = e^{⟨δ,H_{1}⟩}. Thus

∑_{λ_{1}∈L_{1}(π,𝔛)} ϕ_{λ}(H_{1}, a_{2})e^{⟨λ_{1}−λ_{1}^{0},H_{1}⟩} = o(1)

as a_{1} → ∞ in A_{1}^{+} = A^{+}(P). If ε > 0 we can choose R > 0 and a finite subset S of L_{1}(π, 𝔛) so that if ⟨α, H_{1}⟩ ≥ R + εB(H_{1}, H_{1}) when α is a root of A_{1} in 𝔫 then

| ∑_{λ_{1}∈L_{1}(π,𝔛)} ϕ_{λ_{1}}(H_{1}, a_{2})e^{⟨λ_{1}−λ_{1}^{0},H_{1}⟩}| ≤ ε

and

| ∑_{λ_{1}∉S} ϕ_{λ_{1}}(H_{1}, a_{2})e^{⟨λ_{1}−λ_{1}^{0},H_{1}⟩}| ≤ ε.

Then

| ∑_{λ_{1}∈S} ϕ_{λ_{1}}(H_{1}, a_{2})e^{⟨λ_{1}−λ_{1}^{0},H_{1}⟩}| ≤ 2ε.

Lemma A.3.2.3 then implies that

|ϕ_{λ_{1}^{0}}(H_{1}, a_{2})| ≤ 2ε

for all H_{1}. Since ε is arbitrary φ_{λ_{1}^{0}}(H_{1}, a_{2}) = 0.

It remains to show that ρ is essentially tempered. Any K_{1}-finite linear form on 𝔙 is a linear combination of the functionals

θ → θ(m_{1}, φ_{1}, φ_{2})

where m_{1} ∈ M(R) and φ_{1}, φ_{2} are continuous functions on K_{1}. Thus

|f(r(m)θ)| ≤ c(1 + l(m))^{d−1}π_{λ_{0}}(m)Ξ_{M}(m).

A similar inequality is valid for the representation ρ. Set ρ′ = π_{λ^{0}}^{−1} ⊗ ρ. If w ∈ W is K_{1}-finite and f is a K_{1}-finite linear form on W, an inequality

|f(ρ′(m)w)| ≤ c(1 + l(m))^{d−1}Ξ_{M}(m)

is satisfied.

To finish up we have only to prove the following lemma, in which we replace M by G and ρ by π in order to allow the symbols ρ, M, and P to take on a new meaning.

LEMMA 4.10. Suppose that π and π^{∗} are quasi-simple irreducible representations of G(R) on the Banach spaces V and V^{∗} and that there is a nontrivial G(R)-invariant bilinear pairing (u, v) → ⟨u, v⟩ of V × V^{∗} into C. Suppose there is an integer d such that for every K-finite u and v an inequality

|⟨π(g)u, v⟩| ≤ c(1 + l(g))^{d}Ξ_{G}(g)

is satisfied. Then there is a parabolic subgroup P of G over R and a unitary representation ρ of M(R), square-integrable modulo the center, such that π is a constituent of I_{ρ}^{P}.

We start from the expansion (4.1) and show that if λ_{0} ∈ L(π, 𝔛) then ReB(β_{i}, λ_{0}) ≤ 0 for all i. If not, there is a linear combination β = ∑ b_{i}β_{i} with positive coefficients such that ReB(β, λ_{0}) > 0. Choose H_{0} in the Lie algebra of A so that ⟨λ, H_{0}⟩ = B(β, λ) for all λ. Then e^{H_{0}} lies in the interior of A^{+}. Taking Lemma 3.6 and the assumption of the lemma into account, we see that for H in a small neighborhood of H_{0}

∑_{λ∈L(π,𝔛)} p_{λ}(tH)e^{t⟨λ−λ_{0},H⟩} = o(1)

as t → ∞. Applying Lemma A.3.2.3 as before we conclude that p_{λ_{0}}(H) = 0, a contradiction.

Let

E(λ) = {i|Re B(β_{i}, λ) = 0}.

Let E be maximal in the collection of E(λ). P will be defined by demanding that P ⊇ P_{0}, a fixed parabolic subgroup minimal over R, and that D(P) be spanned by D_{0} and {β_{i}|i ∈ E}.

This decided, we turn to the expansion (4.3). There is at least one λ_{1}^{0} in L_{1}(π, 𝔛) with Re λ_{1}^{0} = 0. We fix it and define the function Φ(m) as before. If λ_{1} ∈ L_{1}(π, 𝔛) then Re B(β_{i}, λ_{1}) ≤ 0 for i ∈ E. This allows us to argue as before and to show that the new Φ satisfies (4.8).

It satisfies a much improved form of (4.10). If λ ∈ L(π, 𝔛) has projection λ_{1} in L_{1}(π, 𝔛) and Re λ_{1} = 0 then, by the maximality of E, B(β_{i}, λ) < 0 for i ∉ E. Since the set E(π) is finite there is a μ ∈ D(P_{0}) such that B(β_{i}, μ) = 0 for i ∈ E and B(β_{i}, μ) < 0 for i ∉ E and such that

B(β_{i}, μ) ≥ Re B(β_{i}, λ)

if λ ∈ L(π, 𝔛) and the real part of the projection of λ on D(P) ⊗ C is zero. Theorem 4 of [5] implies that there are an integer d and a constant c such that

(4.14)  π_{δ_{1}}(a)∥Φ(a)∥ ≤ c(1 + l(a))^{d}Ξ_{M}(a)e^{⟨u,H⟩}

for a = e^{H} in A^{+}(P′_{0}), where P′_{0} = P_{0} ∩ M. Using this inequality instead of (4.10) we proceed as before to define ρ. π is then a constituent of I_{ρ}^{P}. Since it follows easily from (4.14) that ρ is square-integrable modulo the center, the lemma is proved.

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