Stable Conjugacy: Definitions and Lemmas*

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.

The difficulties with which trace formula confronts us are manifold. Most of them arise from the noncompactness of the quotient and will not concern us here. Others are primarily arithmetic and occur even when the quotient is compact. To see how they arise, we consider a typical problem.

Suppose G is a quasi-split group over a global field F and G′ is a group obtained from G by an inner twisting. Thus there is an isomorphism φ: G′ → G defined over a finite Galois extension K of F which is such that σ(ψ)ψ^{−1} is inner for all σ ∈ Gal(K/F). Apart from the contributions from the cusps, the trace formula for G, insofar as it is available, expressed the trace as a sum over semi-simple elliptic conjugacy classes, and the trace formula for G′ expresses the trace as a sum over the semi-simple elliptic conjugacy classes of G′(F). The traces of which we speak are those of r(f) or of r′(f′) where f and f′ are suitable functions on G(A(F)) or G′(A(F)) and r and r′ are representations of G(A(F)) and G′(A(F)) on suitable spaces of automorphic forms, which it will be safer not to define precisely.

In its naive form the principle of functoriality suggests, and even affirms, that there is an injection of the set of automorphic representations of G′(A(F)) into the set of automorphic representations of G(A(F)). This is not so, and if we attempt to prove it by following the standard paradigm ([6], §16), we will discover why. We must show that the traces of r(f) and r(f′) are equal for suitable test functions. The best procedure is to consider first the contributions from the elliptic conjugacy classes, and then, confidence gained, to pass to the cuspidal terms or, our misapprehensions revealed, to modify our expectations.

To compare the two traces one considers the two trace formulae and compares them term-by-term. If γ′ is a semi-simple element in G′(F) then the conjugacy class of ψ(γ′) is defined over F because σ(ψ(γ′)) = σ(ψ)ψ^{−1}(ψ(γ′)) and σ(ψ)ψ^{−1} is inner. A theorem of Steinberg [11] then assures us that the conjugacy class of ψ(γ′) in G(F̄) contains an element γ in G(F). If conjugacy within G(F̄), called stable conjugacy, is the same as conjugacy within G(F), and if conjugacy within G′(F̄) is the same as conjugacy within G′(F) then we obtain an injection of the elliptic conjugacy classes of G′(F) into the elliptic conjugacy classes of G(F) and can hope, by means of a supplementary study of the local harmonic analysis, to show that the terms corresponding to associated classes {γ} and {γ′} are equal, and that, for the functions being tested, the term

* Appeared in Can. J. Math., Vol. XXXI, No. 4, 1979, 700–725. of the trace formula corresponding to a class {γ} not associated to a conjugacy class in G′(F) is zero. This is the method used for GL(2) [6] and GL(3) [4], and one expects that it will eventually deal with GL(n).

For other groups stable conjugacy will be different from conjugacy, but at first glance this appears to be no serious obstacle. One should simply group together those terms corresponding to the conjugacy classes lying within a stable conjugacy class, obtaining thereby sums over stable conjugacy classes which can then be compared term-by-term. But the comparison of terms has to be carried out by an analysis of local orbital integrals to which the sum over a global stable conjugacy class is not directly amenable. Indeed the two terms to be compared are unlikely to be equal. A further adelic stabilization is necessary, but this can only be done by adding terms not present in the trace formula, and so they must be again subtracted, as an error term. The fully stabilized trace formulae will probably be amenable to comparison by a local study of orbital integrals, but a supplementary analysis of the error term is now necessary.

It may be possible to effect this by a procedure which may strike the more prosaic of our readers as extravagant. Regarding the stabilized trace formula as basic, we try to express the error term as a sum of stabilized trace formulae for lower-dimensional groups H, whose representation theory is related to that of G by the principle of functoriality.

All this will take time, and the efforts of more than one. My purpose here is simply to give the definitions of the groups H which intervene in the error term, together with their elementary properties. The definitions emerged from a close examination of a special case, SL(2), for which the procedure outlined has been carried out in detail ([7], [14]).

The groups H can also be introduced locally, where their purpose is to reduce the harmonic analysis of invariant distributions to the analysis of stably invariant distributions, and the local problems must be solved previously to, or simultaneously with, the global problems. For SL(2) they are either easy or had already been treated. For other groups this is not so, and even over the field of real numbers they are novel and difficult, but are yielding to the efforts of Shelstad ([12], [13]), whose work does much to dispel our doubts about the value of the definitions below.

At first F can be any field of characteristic 0 and G a reductive group over it. Let T = T^{G} be a Cartan subgroup of G. Let 𝔄(T) or 𝔄(T, F) be the set of all g in G(F̄) for which T′ = g^{−1}T g and the morphism t → t′ = g^{−1}tg are both defined over F and let

𝔇(T, F) = 𝔇(T) = T (F̄)\𝔄(T)/G(F) .

An element g in G(F̄) lies in 𝔄(T) if and only if a_{σ} = σ(g)g^{−1} lies in T (F̄) for all σ ∈ Gal(F̄/F). The collection {a_{σ} | σ ∈ Gal(F̄/F)} defines a cohomology class in H^{1}(F, T) and the map g → {a_{σ}} yields an injection

𝔇(T) ↪ H^{1}(F, T) .

The image is the kernel of

H^{1}(F, T) → H^{1}(F, G)

and is not always a group. If T′ = g^{−1}T g with g ∈ 𝔄(T) then T and T′ are said to be stably conjugate. The set 𝔇(T) parametrizes the conjugacy classes within the stable conjugacy class of T.

If G_{sc} is the simply-connected covering group of the derived group of G and T_{sc} the inverse image of T in G_{sc} then

𝔇(T_{sc}) → 𝔇(T)

is surjective. We define 𝔈(T) or 𝔈(T, F) to be the image of H^{1}(F, T_{sc}) in H^{1}(F, T). It is a group and 𝔇(T) is a subset of it. If F is local and non-archimedean then H^{1}(F, G_{sc}) = {1} and 𝔇(T) = 𝔈(T).

Let X_{∗}(T) and X_{∗}(T_{sc}) be the lattices of coweights of T and T_{sc}. X_{∗}(T_{sc}) may be identified with the sublattice of X_{∗}(T) generated by the coroots. If E is a local field and K a large but finite Galois extension then, by the Tate-Nakayama theory, 𝔈(T) is canonically isomorphic to the quotient of

{λ ∈ X_{∗}(T_{sc})| ∑_{Gal(K/F)} ω_{T/G}(σ)λ = 0}

by

{λ ∈ X_{∗}(T_{sc})|λ = ∑_{Gal(K/F)} ω_{T/G}(σ)μ(σ) − μ(σ), μ(σ) ∈ X_{∗}(T)} .

Here ω_{T/G}(σ) is the natural action of σ on X_{∗}(T). If F is any field we let κ be a homomorphism of X_{∗}(T_{sc}) into C^{×} which is 1 on the second of these modules.

I am now going to associate to the pair (T, κ) = (T^{G}, κ) a quasi-split group H over F, and a family {T^{H}, φ)} where T^{H} is a Cartan subgroup of H, and φ : T^{G} → T^{H} is an isomorphism over F. If (T_{1}^{H}, φ_{1}) and (T_{2}^{H}, φ_{2}) are two pairs in the family then there is an h in 𝔄(T_{1}^{H}) for which

T_{2}^{H} = h^{−1}T_{1}^{H}h

and

φ_{2}(t) = h^{−1}φ_{1}(t)h .

To define H one needs the associate group of [10], which following Borel [1] I denote ^{L}G. Its connected component ^{L}G^{0} is furnished with a distinguished Borel subgroup ^{L}B^{0} and a distinguished Cartan subgroup LT^{0}. The group ^{L}T^{0} is contained in ^{L}B^{0}. To define ^{L}G^{0} concretely we need to choose an isomorphism ψ of G with a quasi-split G_{1}. G_{1} is defined over F, but ψ is only defined over F̄. We also need to choose a Borel subgroup B^{G_{1}} and a Cartan subgroup T^{G_{1}} lying in B^{G_{1}}, both groups being defined over F. Choose g_{1} in G_{1}(F̄) so that

ψ′ = ad g_{1} ∘ ψ

takes T^{G} to T^{G_{1}}. Then g_{1} is determined up to left multiplication with an element of Norm (T^{G_{1}}).

ψ′ also yields an isomorphism

ψ′ : X_{∗}(T^{G}) → X_{∗}(T^{G_{1}})

and by the construction of ^{L}G

X_{∗}(T^{G_{1}}) = X^{∗}(^{L}T^{0})

if X^{∗}(^{L}T^{0}) is the lattice of weights of ^{L}T^{0}. Define κ′ by

κ′(ψ′(λ)) = κ(λ)

and let ^{L}H^{0} be the connected subgroup of ^{L}G^{0} generated by ^{L}T^{0} and the one-parameter root groups U_{α^{∨}} associated to α^{∨} with

κ′(α^{∨}) = 0 .

LH^{0} is furnished with a distinguished Cartan subgroup, namely ^{L}T^{0}, and a distinguished Borel subgroup

LH^{0} ∩^{L}B^{0} .
We transfer the operators ω_{T/G}(σ) to X_{∗}(^{L}T^{0}) and write
ω_{T/G}(σ) = ω^{1}(σ)ω^{2}(σ)

where ω^{2}(σ) is given by an element of the Weyl group of ^{L}T^{0} in ^{L}H^{0} and ω^{1}(σ) leaves the set of positive roots of ^{L}T^{0} in ^{L}H^{0} invariant. If for each simple root X_{α^{∨}} we choose the X_{α^{∨}} ≠ 0 in the Lie algebra of U_{α^{∨}} used in the definition of ^{L}G we may associate to ω^{1}(σ) a unique automorphism of ^{L}H^{0}, again denoted by ω^{1}(σ), with the following properties:

ω^{1}(σ)λ(ω^{1}(σ)t) = λ(t), λ ∈ X^{∗}(^{L}T^{0}), t ∈ ^{L}(T^{0}); ω^{1}(σ)X_{α^{∨}} = X_{ω^{1}(σ)α^{∨}} .

If we choose the Galois extension K of F sufficiently large, then σ → ω^{1}(σ) is a homomorphism of Gal(K/F) into the group of automorphisms of ^{L}H^{0}. By means of W_{K/F} → Gal(K/F) we let the Weil group act and form the semi-direct product

LH = ^{L}H^{0} × W_{K/F} .

As in [10] we associate to ^{L}H^{0}, ^{L}H^{0} ∩^{L}B^{0}, ^{L}T^{0}, {X_{α^{∨}}}, and {ω^{1}(σ)} a quasi-split group H, furnished with a Cartan subgroup T_{0}^{H}, and a Borel subgroup B^{H}, all defined over F, so that

X_{∗}(T_{0}^{H}) = X^{∗}(^{L}T^{0})

and

ω_{T_{0}^{H}/H}(σ) = ω^{1}(σ) .

The concatenation of

X_{∗}(T^{G}) → X_{∗}(T^{G_{1}}) = X^{∗}(^{L}T^{0}) = X_{∗}(T_{0}^{H})

yields an isomorphism

φ′ : X_{∗}(T^{G}) → X_{∗}(T_{0}^{H})

and hence an isomorphism

φ′ : T^{G} → T_{0}^{H} .

However φ′ is not defined over F. By a theorem of Steinberg [11] there is at least one Cartan subgroup T^{H} of H over F given by

T^{H} = h^{−1}T_{0}^{H}h, h ∈ H(F̄)

such that the composition

φ : T^{G} ⟶^{φ′} T_{0}^{H} ⟶^{ad h^{−1}} T^{H}

is defined over F. The pairs (T^{H}, φ) obtained in this way form the family I mentioned. I forego for now a close examination of the manner in which H and the family {(T^{H}, φ)} depend on the choices required for their construction, merely stressing that once ψ is fixed one must still choose g_{1}; hence H = H(T, κ, g_{1}). The triple (T_{2}, κ_{2}, g_{2}) will be called a companion to (T_{1}, κ_{1}, g_{1}) if the associated homomorphisms κ′_{1}, κ′_{2} of X_{∗}(T_{sc}^{G_{1}}) to C^{×} and the associated operators ω_{1}^{1}(σ), ω_{2}^{1}(σ) on X^{∗}(^{L}T^{0}) are equal. Then H(T_{1}, κ_{1}, g_{1}) and H(T_{2}, κ_{2}, g_{2}) are canonically isomorphic.

By its construction we have an imbedding ξ : ^{L}H^{0} ↪ ^{L}G^{0}. In order to bring the principle of functoriality in the dual group into play we need to extend it to an imbedding ξ : ^{L}H ↪ ^{L}G which commutes with the projections on W_{K/F}. This is not always possible, but it is possible in sufficiently many cases that the groups H can be used for the purpose for which they were intended, the study of L-indistinguishability.

Proposition 1. Suppose F is a global or a local field and the center of ^{L}G^{0} is connected. Then the imbedding ξ : ^{L}H^{0} ↪ ^{L}G^{0} extends to an imbedding ξ : ^{L}H ↪ ^{L}G which commutes with the projections on W_{K/F}.

I shall argue by induction on the dimension of ^{L}G^{0}. The statement is certainly clear if the dimension is zero. The center of ^{L}G^{0} is connected if and only if the lattice X_{∗}(T_{sc}^{G_{1}}) is primitive in X_{∗}(T^{G_{1}}), that is, if any rational linear combination of the coroots which is a coweight is in fact an integral linear combination.

There is an integer m such that the image of X_{∗}(T_{sc}^{G_{1}}) under λ → κ′(λ)^{m} is torsion free. Choose independent homomorphisms η_{1}, …, η_{s} of X_{∗}(T_{sc}^{G_{1}}) into Z and complex numbers ζ_{1}, …, ζ_{s} so that

κ′(λ)^{m} = ∏_{i=1}^{s} ζ_{i}^{η_{i}(λ)} .

Let 𝔛 be the set of roots α^{∨} for which the first non-zero η_{i}(α^{∨}) is positive and 𝔛_{0} ⊆ 𝔛 the set of roots α^{∨} for which all η_{i}(α^{∨}) are zero. The group ^{L}P^{0} generated by ^{L}T^{0} and the one-parameter root groups U_{α^{∨}} for which α^{∨} ∈ 𝔛 is a parabolic subgroup of ^{L}G^{0} and a Levi factor ^{L}M^{0} is generated by ^{L}T^{0} and the U_{α^{∨}} with α^{∨} ∈ 𝔛_{0}. Since 𝔛 is invariant under ω_{T/G}(σ), σ ∈ 𝔊(F̄/F), the normalizer ^{L}P of ^{L}P^{0} in ^{L}G is a parabolic subgroup of ^{L}G in the sense of [10]. The normalizer ^{L}M of ^{L}M^{0} in ^{L}G is itself an L-group. Certainly LH^{0} ⊆ ^{L}M^{0}. Hence, if

dim(^{L}M^{0}) < dim(^{L}G^{0})

and the center of ^{L}M^{0} is connected, we may apply induction. However, the center is connected because it is defined by

α^{∨}(t) = 0

for each simple root α^{∨} of ^{L}T^{0} in ^{L}M^{0}. These simple roots generate a primitive lattice in X^{∗}(^{L}T^{0}) = X_{∗}(T^{G_{1}}).

The upshot of the preceding analysis is that we need only consider κ′ that are of finite order. Some preparation is necessary.

Lemma 2. Suppose R is an indecomposable, reduced root system and D a subset of R with the following two properties.

(i) If α^{∨}, β^{∨} lie in D then α^{∨} − β^{∨} is not a root.

(ii) Every root of R is an integral linear combination of the elements of D. Then D is either a base of R or a base together with the negative of the corresponding highest root.

This lemma is implicit in [3]. As one expects, a proof can also be extracted from the thesaurus of Bourbaki [2]. If α^{∨} ≠ β^{∨} both lie in D then certainly (α^{∨}, β^{∨}) ≦ 0. One defines a Coxeter matrix (m_{α^{∨}, β^{∨}}), α^{∨}, β^{∨} in D, by taking

π − \frac{π}{m_{α^{∨}, β^{∨}}}

to be the angle between α^{∨}, β^{∨}. Let {e_{α^{∨}}} be the standard basis of

E = ⊕_{α^{∨}∈D} R

and define an inner product B( · , · ) on E by

B(e_{α^{∨}}, e_{β^{∨}}) = cos (π − \frac{π}{m_{α^{∨}, β^{∨}}} ) = \frac{(α^{∨}, β^{∨})}{(α^{∨}, α^{∨})^{1/2}(β^{∨}, β^{∨})^{1/2}}

Since

B(∑ a(α^{∨})e_{α^{∨}}, ∑ a(β^{∨})e_{β^{∨}}) = (∑ \frac{a(α^{∨})}{(α^{∨}, α^{∨})^{1/2}} α^{∨}, ∑ \frac{b(β^{∨})}{(β^{∨}, β^{∨})^{1/2}} β^{∨})

the inner product is positive semi-definite.

Suppose the form is definite or, what amounts to the same thing, that the roots α1^{∨}, …, αl^{∨} of D are linearly independent. If 1 ≤ k ≤ l let R_{k} be the set of the roots in the real linear span V_{k} of {α1^{∨}, …, αk^{∨}}. R_{k} is a root system and every element of R_{k} is an integral linear combination of α1^{∨}, …, αk^{∨}. We show by induction that α1^{∨}, …, αk^{∨} is a base of R_{k} or, more precisely, that if we define an order on V_{k} by

∑_{i=1}^{k} a_{i}αi^{∨} > 0

if and only if the last non-zero a_{i} is positive, then α1^{∨}, …, αk^{∨} are the minimal elements of R_{k} with respect to this order. This is clear for k = 1.

The induction assumption will be that the assertion is true for a given k regardless of the initial numeration of the roots in D. If k < l it is clear that the minimal elements in R_{k+1} are then α1^{∨}, …, αk^{∨} together with some β^{∨}. Moreover

αk+1^{∨} = a_{1}α1^{∨} + ⋯ + a_{k}αk^{∨} + aβ^{∨}

with a_{1}, …, a_{k} and a integral and non-negative. Solving for β^{∨} we see that a = 1; so we write

ak+1^{∨} = γ^{∨} + β^{∨}

with

γ^{∨} = a_{1}α1^{∨} + ⋯ + a_{k}αk^{∨} .

We suppose that γ^{∨} ≠ 0 and derive a contradiction. Since αk+1^{∨} − αj^{∨}, 1 ≦ j ≦ k, is not a root, γ^{∨} = αk+1^{∨} is. This implies that k > 1, for if k were 1 then a_{1} would be 1 and α2^{∨} − a1^{∨} = β^{∨} would be a root. We observe that

(αk+1^{∨}, αk+1^{∨}) = ∑_{i=1}^{k} a_{i}(αk+1^{∨}, αi^{∨}) + (αk+1^{∨}, β^{∨}) ≦ (αk+1^{∨}, β^{∨}) .

If (αk+1^{∨}, αk+1^{∨}) were equal to (β^{∨}, β^{∨}) we would conclude from this inequality that αk+1^{∨} = β^{∨}. However, this cannot be so, for γ^{∨} is supposed not to be zero. We infer, therefore, from the above inequality combined with the Schwarz inequality that

(αk+1^{∨}, αk+1^{∨}) < (β^{∨}, β^{∨}) .

Since the rank of R is greater than 2 and R is irreducible, we must have

2(αk+1^{∨}, αk+1^{∨}) = (β^{∨}, β^{∨}) .

The geometrical situation is:

Stable conjugacy—definitions and lemmas 8

β^{∨}

αk+1^{∨}

γ^{∨}

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . .

.

.

.

.

.

.

.

.

.

Hence

(αk+1^{∨}, αk+1^{∨}) = (αk+1^{∨}, β^{∨}) .

Consequently

a_{i}(αk+1^{∨}, αi^{∨}) = 0

for 1 ≦ i ≦ k and

a_{i}(γ^{∨}, αi^{∨}) = −a_{i}(β^{∨}, αi^{∨}) ≧ 0 .

However, our initial assumption that our assertion is valid at the kth stage regardless of the initial numeration implies that each a_{i} > 0. Moreover, since γ^{∨} ≠ 0 there is one root αi^{∨} with (β^{∨}, αi^{∨}) < 0. Then

(β^{∨}, αi^{∨}) = −(αi^{∨}, αi^{∨})

if αi^{∨} is short, and

(β^{∨}, αi^{∨}) = −\begin{matrix}1 \\ – \\ 2 \end{matrix} (αi^{∨}, αi^{∨})

if αi^{∨} is long. However, γ^{∨} is short. Therefore,

(γ^{∨}, αi^{∨}) = \begin{matrix}1 \\ – \\ 2 \end{matrix} (αi^{∨}, αi^{∨})

regardless of the length of αi^{∨}. We conclude that αi^{∨} is long and that

0 < (αk+1^{∨}, β^{∨}) ≦ a_{i}(αi^{∨}, β^{∨}) + (β^{∨}, β^{∨}) = (1 − a_{i}/2)(β^{∨}, β^{∨}) .

It follows that a_{i} = 1.

There is another root αj^{∨}, 1 ≦ j ≦ k, with (αj^{∨}, αi^{∨}) < 0. Since the Dynkin diagram of R_{k+1} contains no cycles, (αj^{∨}, β^{∨}) = 0 and

0 = (αk+1^{∨}, αi^{∨}) ≦ a_{j}(αj^{∨}, αi^{∨}) + a_{i}(αi^{∨}, αi^{∨}) + (αi^{∨}, β^{∨}) = a_{j}(αj^{∨}, αi^{∨}) + \begin{matrix}1 \\ – \\ 2 \end{matrix} (αi^{∨}, αi^{∨}) .

Since

(αj^{∨}, αi^{∨}) = −\begin{matrix}1 \\ – \\ 2 \end{matrix} (αi^{∨}, αi^{∨})

no matter whether αj^{∨} is long or short, we conclude that a_{j} = 1.

Suppose we have a path leading out from β^{∨} in the Dynkin diagram with at least three vertices besides β^{∨} in it. Suppose moreover that we have shown that the coefficient a_{m} of αm^{∨} is 1 for all vertices of the path except perhaps the last and that all vertices except perhaps for the last two are long. Let the last vertices

be αu^{∨}, αv^{∨}, αw^{∨}. Then

0 = (αk+1^{∨}, αv^{∨}) ≦ a_{w}(αw^{∨}, αv^{∨}) + (αv^{∨}, αv^{∨}) + (αu^{∨}, αv^{∨}) .

If αv^{∨} were short, then

(αv^{∨}, αv^{∨}) + (αu^{∨}, αv^{∨}) = 0

and the contradiction

0 ≦ (αw^{∨}, αv^{∨}) < 0

results. Thus αv^{∨} is long and

0 ≦ a_{w}(αw^{∨}, αv^{∨}) + \begin{matrix}1 \\ – \\ 2 \end{matrix} (αv^{∨}, αv^{∨}) .

We conclude that a_{w} = 1. By induction all the coefficients are 1. Then if αm^{∨} is an extreme point of the Dynkin diagram we infer from Cor. 3 to Prop. 19 of Chap. VI of [2] that αk+1^{∨} = αm^{∨} is a root. This is a contradiction.

We must still prove the lemma when the form B( · , · ) is degenerate. In general the collection D may be partitioned into subsets D_{1}, …, D_{r} corresponding to the connected components of the graph of the Coxeter matrix we have introduced. If V_{i} is the space over R spanned by D_{i} then V_{1}, …, V_{r} are mutually orthogonal and any root in V_{i} is a linear combination of elements of D and hence of D_{i}. Let R_{i} = R ∩ V_{i}.

Suppose next that the graph of our Coxeter matrix is connected but that the form B( · , · ) is degenerate. It is a consequence of Th. 4 of Chap. VI of [2] that the Coxeter matrix (m_{α^{∨},β^{∨}}) is defined by the Weyl group of a completed Dynkin diagram. Hence there is only one relation between the roots α1^{∨}, …, αl^{∨} of D and with a suitable numeration it is

( ∑_{1}^{l−1} a_{i}αi^{∨}) + αl^{∨} = 0 .

We may remove αl^{∨} from D without destroying either of the two properties demanded of it. Since α1^{∨}, …, αl−1^{∨} are linearly independent, the previous discussion implies that they form a base of R.

Returning to the general case, we see that we can select from any of the D_{i} a base for the corresponding R_{i}. Putting these bases together, we find a collection that satisfies (i) and (ii) of the lemma and is in addition linearly independent. We conclude from the first part of the proof that it is a base. Since R is indecomposable, we infer that r = 1. The lemma is now proved.

We return to the proof of Proposition 1, supposing now that κ′ is of finite order m. Let

ζ = e^{2πi/m} .

Define 𝔜_{k} for 0 ≦ k < m by

𝔜_{k} = {α^{∨}|κ′(α^{∨}) = ζ^{k}} .

Let 𝔛_{0} be the set of simple roots of ^{L}T^{0} in ^{L}H^{0} with respect to ^{L}H^{0} ∩^{L} B^{0}. If α^{∨}, β^{∨} lie in 𝔜_{k} write α^{∨} < β^{n} if β^{∨} − α^{∨} is an integral linear combination of roots of 𝔛_{0} with non-negative coefficients. Let ℨ_{k} consist of those elements of 𝔜_{k} which are not integral linear combinations of elements in

⋃_{0}^{k−1} 𝔜_{j}

and let 𝔛_{k}, k ≧ 1 consist of the minimal elements in ℨ_{k}. Clearly 𝔛_{0}, 𝔛_{1}, …, 𝔛_{m−1} are disjoint and span X_{∗}(T_{sc}^{G_{1}}) over Z. Moreover, each of these sets is invariant under ω^{1}(σ), σ ∈ 𝔊(F̄/F). If α^{∨} ∈ 𝔛_{k}, β^{∨} ∈ 𝔛_{j} and j ≦ k then α^{∨} − β^{∨} is not a root. Suppose otherwise. If 0 < j < k then α^{∨} − β^{∨} ∈ 𝔜_{k−j}, and so α^{∨} is not in ℨ_{k}. If j = 0 then α^{∨} − β^{∨} ∈ ℨ_{k} and α^{∨} is not minimal. Lemma 2 may be applied to

D = ⋃_{0}^{m−1} 𝔛_{j}

when ^{L}G^{0} is simple, and yields an important tool for the study of ^{L}H^{0}. The Dynkin diagram of D together with the action ω^{1}(σ) of 𝔊(F̄/F) on it will be called the diagram of (T, κ) or of (T, κ, g_{1}). Each vertex is labelled with an integer k, 0 ≦ k < m.

Choose an X_{α^{∨}} for each α^{∨} in 𝔛_{0}. We denote a typical element of W_{K/F} by w and its image in Gal(K/F) by σ. For each w we may choose ξ′(w) in ^{L}G so that ξ′(w) project to w in W_{K/F} and so that

\begin{matrix}ξ′(w)tξ′(w)^{−1} = ω^{1}(σ)(t), t ∈^{L} T^{0}, \\ ξ′(w)X_{α^{∨}}ξ′(w)^{−1} = X_{ω^{1}(σ)α^{∨}} . \end{matrix}

ξ′(w) is not uniquely determined, but we may modify it only by left multiplication with elements from Z, the center of ^{L}H^{0}. Thus

ξ′(w_{1})ξ′(w_{2}) = a_{w_{1},w_{2}}ξ′(w_{1}w_{2})

with a_{_{1},w_{2}} ∈ Z. Clearly {a_{w_{1}w_{2}}} defines a 2-cocycle of W_{K/F} with values in Z. Since, as a first try, we can even take ξ′(w) to depend only on σ, it is continuous. Our problem is to show that it is trivial, that

a_{w_{1},w_{2}} = b(w_{1})ω^{−1}(σ_{1})(b(w_{2}))b^{−1}(w_{1}w_{2})

with w → b(w) ∈ Z(K) continuous.

The first step is to show that if we take ξ′(w) to depend only on σ so that {a_{w_{1},w_{2}}} = {a_{σ_{1},σ_{2}}} is a cocycle of Gal(K/F) with values in Z(K) then it is trivial modulo Z^{0}(K), Z^{0} being the connected component of the identity in Z. Since the center of ^{L}G^{0} is connected, we may divide by it and assume that ^{L}G^{0} is adjoint. Then an application of Shapiro’s lemma allows us to assume that ^{L}G^{0} is simple.

If the diagram of (T, κ) is ordinary and not extended then the roots in 𝔛_{0} generate a primitive lattice in

X_{∗}(T_{sc}^{G_{1}})

and Z is connected. For now we take the diagram to be extended. We write the vertices of it as α^{∨}, α1^{∨}, …, αl^{∨} and the one relation as

α^{∨} + ∑_{i=1}^{l} a_{i}αi^{∨} = 0 .

Here the a_{i} are positive integers. It is clear that Z will be connected unless α^{∨} ∈ 𝔛_{0} and

g.c.d.{a_{i}|αi^{∨} ∉ 𝔛_{0}} > 1 .

We shall examine the possible diagrams individually. Given ^{L}G^{0}, ^{L}B^{0}, ^{L}T^{0} and {X_{α^{∨}}|α^{∨} simple}, we forget their origins in G but take the group A(^{L}G^{0},^{L}B^{0},^{L} T^{0}, {X_{α^{∨}}}) as on p. 4 of [10] and build the semi-direct product

LG′ = ^{L}G^{0} × A(^{L}G^{0},^{L}B^{0},^{L} T^{0}, {X_{α^{∨}}}) .

Recall that A(^{L}G^{0},^{L}B^{0},^{L} T^{0}, {X_{α^{∨}}}) is the group of automorphisms of ^{L}G^{0} which leave ^{L}B^{0} and ^{L}T^{0} invariant and permute the X_{α^{∨}} amongst themselves. In addition we consider an extended Dynkin diagram of ^{L}T^{0} in ^{L}G^{0} with vertices D = {α^{∨}, α1^{∨}, …, αl^{∨}} where α1^{∨}, …, αl^{∨} are the simple positive roots with respect to another ordering of the roots than that defining ^{L}B^{0} and a subset 𝔛_{0} of D whose elements are positive with respect to the original order. We suppose that α^{∨} ∈ 𝔛_{0} and that

g.c.d.{a_{i}|αi^{∨} ∉ 𝔛_{0}} > 1 .

We let A be the subgroup of Norm_{^{L}G′} (^{L}T^{0})/^{L}T^{0} formed by those ω that leave D and 𝔛_{0} invariant. We assign to each ω a representative ϵ(ω) in Norm_{^{L}G′} (^{L}T^{0}) so that

ϵ(ω)tϵ(ω)^{−1} = ω(t), t ∈ ^{L}T^{0},
ϵ(ω)X_{α^{∨}}ϵ(ω)^{−1} = X_{ωα^{∨}}, α^{∨} ∈ 𝔛_{0} .

Then

ϵ(ω_{1})ϵ(ω_{2}) = a_{ω_{1},ω_{2}}ϵ(ω_{1}ω_{2}) .

{a_{ω_{1},ω_{2}}} is a cocycle of A with coefficients in

Z = {t ∈ ^{L}T^{0}|α^{∨}(t) = 1 for α^{∨} ∈ 𝔛_{0}} .

We shall show that this cocycle is trivial modulo Z^{0}. It is enough to show that its restriction to a Sylow subgroup A_{p} of A is trivial for each p.

We now check this by examining the possible diagrams one-by-one, excluding those for which A = {1} or Z is connected because the assertion is then trivial. Diagrams of type A_{l} do not appear because all the a_{i} are then 1, and for diagrams of type E_{8}, F_{4}, G_{2} the group A is {1}.

1) B_{l}, l ≧ 3.

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α^{∨}

α1^{∨}

α2^{∨} αl−1^{∨} αl^{∨}

•

•

• o • •

α^{∨} + α1^{∨} + 2α2^{∨} + ⋯ + 2αl^{∨} = 0

Here α^{∨} and α1^{∨} must belong to 𝔛_{0} and A = {1, ω} where ω interchanges α^{∨} and α1^{∨} and fixes α2^{∨}, …, αl^{∨}. With the standard representation of the root system of type B_{l}

α1^{∨} = x_{1} − x_{2}, α2^{∨} = x_{2} − x_{3}, …, αl−1^{∨} = x_{l−1} − x_{l}, αl^{∨} − x_{l}, α^{∨} = −x_{1} − x_{2}.

Thus ω is the reflection with respect to β^{∨} = x_{1}. If β is the corresponding root then β = 2x_{1} and ⟨β, λ^{∨}⟩ is even for all λ^{∨} ∈ X^{∗}(^{L}T^{0}). If φ_{β^{∨}} is the map of SL(2) into ^{L}G^{0} given in the usual way once X_{β^{∨}} is chosen, we may take

ϵ(ω) = φ_{β^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & −x \\ x^{−1} & 0 \end{matrix} & ) \\ & & ) \end{aligned}

then

ϵ(ω)X_{γ^{∨}}ϵ(ω)^{−1} = X_{γ^{∨}} γ^{∨} ∈ 𝔛_{0} γ^{∨} ≠ α^{∨}, α1^{∨}

and for a suitable choice of x

ϵ(ω)X_{α_{1}^{∨}}ϵ(ω)^{−1} = X_{α^{∨}}.

Since

λ^{∨}(ϵ(ω)^{2}) = (−1)^{⟨β,λ^{∨}⟩} = 1

we conclude that ϵ(ω)^{2} = 1 and that

ϵ(ω)X_{α^{∨}}ϵ(ω)^{−1} = X_{α_{1}^{∨}}.

The cocycle therefore splits.

2) C_{l}, 1 ≧ 2.

⋯ ........ ....... ........ ....... ............. .. ........ ....... o α^∨_{l−1} o o o o α^∨  α^∨_1  α^∨_l

α^{∨} + 2α1^{∨} + ⋯ + 2αl−1^{∨} + αl^{∨} = 0

Here α^{∨} and α1^{∨} must belong to 𝔛_{0} and A = {1, ω} where ω reflects the diagram in its center. We realize LG^{0} as usual as the symplectic group in 2l variables modulo its center. With the usual representation of the

roots α^{∨} = −2x_{1}, α1^{∨} = x_{1} − x_{2}, …, αl−1^{∨} = x_{l−1} − x_{l}, αl^{∨} = 2x_{l}.

Suppose first that l = 2k is even. Then ω fixes the roots

x_{1} − x_{2k},  x_{2} − x_{2k−1}, …,  x_{k} − x_{k+1}.

The only roots orthogonal to all of these are

β1^{∨} = x_{1} + x_{2k},  β2^{∨} = x_{2} + x_{2k−1}, …,  x_{k} + x_{k+1}

and if

δ(ω) = ∏_{i=1}^{k} φ_{β_{j}^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned}

then δ(ω) is a representtive of ω. The only root of D fixed by ω is α_{k^{∨}} and

\begin{matrix}Ad φ_{β_{j}^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned}: X_{α_{k}^{∨}} → X_{α_{k}^{∨}}, j ≠ k, \\ Ad φ_{β_{k}^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned}: X_{α_{k}^{∨}} → −X_{α_{k}^{∨}}. \end{matrix}

Thus

Ad δ(ω): X_{α_{k}^{∨}} → −X_{α_{k}^{∨}}.

Moreover

λ^{∨}(δ(ω)^{2}) = (−1)^{⟨∑ β_{i},λ^{∨}⟩}.

Since

⟨∑ β_{i}, λ^{∨}⟩ = λ1^{∨} + ⋯ + λl^{∨},

if λ1^{∨}, …, λl^{∨} are the coordinates of λ^{∨}, it is always even and δ(ω)^{2} = 1. Thus if β^{∨} ≠ ωβ^{∨} lies in D and

Ad δ(ω): X_{β^{∨}} → c(β^{∨})X_{ωβ^{∨}}

then

Ad δ(ω): X_{ωβ^{∨}} → c(β^{∨})^{−1}X_{β^{∨}}.

Define t in ^{L}T^{0} by

β^{∨}(t) = \{ \begin{matrix}1 & β^{∨} ∈ D, β^{∨} ∉ 𝔛_{0} \\ c(β^{∨})^{−1} & β^{∨} ∈ 𝔛_{0}. \end{matrix}

These demands are consistent and we may take ϵ(ω) = δ(ω)t. Then

ϵ(ω)^{2} = 1.

If l = 2k + 1 is odd we take

β1^{∨} = x_{1} + x_{2l+1}, β2^{∨} = x_{2} + x_{2l}, …, βk^{∨} = x_{k} + x_{k+2}

and argue as before with

δ(ω) = ∏_{i−1}^{k} φ_{β_{i}^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned}

Since ω fixes no root of 𝔛_{0} or even of D the argument is easier.

3) D_{l}, l ≧ 4.

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•

•  •  •

•

•

•

α^{∨}

α1^{∨}

α2^{∨}

αl^{∨}

αl−1^{∨}

αl−2^{∨}

α^{∨} + α1^{∨} + 2α2^{∨} ⋯ + 2αl−2^{∨} + αl−1^{∨} + αl^{∨} = 0

𝔛_{0} must contain α^{∨}, α1^{∨}, αl−1^{∨}, and αl^{∨}. If l > 4 then A is at most of order 8; if l = 4 then A_{2} is again at most of order 8 while A_{3} is {1} or Z_{3}. Since Z/Z^{0} is of order two we need in any case only consider the cocycle on A_{2}. With the usual representation of the roots

α^{∨} = −x_{1} − x_{2}, α1^{∨} = x_{1} − x_{2}, α2^{∨} = x_{2} − x_{3}, …, αl−1^{∨} = x_{l−1} − x_{1}, αl^{∨} = x_{l−1} + x_{l}.

If l = 2k then the roots

x_{2} − x_{l−1}, x_{3} − x_{l−2}, …, x_{k} − x_{k+1}

are fixed by all the elements of A_{2}. The roots orthogonal to this set are

β^{∨} = x_{1} − x_{l}, β1^{∨} = x_{1} + x_{l}, β2^{∨} = x_{2} + x_{l−1}, …, βk^{∨} = x_{k} + x_{k+1}.

The group A_{2} is generated by ω_{1}, ω_{2}, ω_{3} where ω_{1} interchanges α1^{∨} and α^{∨} but fixes the other roots of D, while ω_{2} interchanges αl−1^{∨} and αl^{∨} and fixes the other roots. Finally ω_{3} interchanges α^{∨} with αl^{∨}, α1^{∨} with αl−1^{∨}, α2^{∨} with αl−2^{∨}, and so on, and fixes αk^{∨}. The defining relations are

ω_{1}^{2} = ω_{2}^{2} = ω_{3}^{2} = 1
ω_{1}ω_{2} = ω_{2}ω_{1}
ω_{3}ω_{1} = ω_{2}ω_{3}.

By its construction there is in ^{L}G′ an element δ_{2} normalizing ^{L}T^{0}, representing ω_{2}, and satisfying

δ_{2}X_{α_{i}^{∨}}δ_{2}^{−1} = X_{α_{i}^{∨}} 1 ≦ i ≦ l − 2

δ_{2}X_{α_{l−1}^{∨}}δ2^{∨} = X_{α_{l}^{∨}}

δ_{2}X_{α_{l}^{∨}}δ_{2}^{−1} = X_{α_{”i−1}^{∨}}.

In addition

δ_{2}^{2} = 1 .

We recall a fact which is verified in [10].

Lemma 3. Suppose ϵ ∈ A(^{L}G^{0},^{L}B^{0},^{L}T^{0}, {X_{α^{∨}}}) and β^{∨} is a root fixed by ϵ. Let β^{∨} = ∑ a(α)α^{∨} be its expression as a sum of simple roots and let a be the sum over the pairs {α^{∨}, ϵα^{∨}} with α^{∨} ≠ ϵα^{∨} and (α^{∨}, ϵα^{∨}) ≠ 0 of a(α) = a(ϵα). Then

ϵ(X_{β^{∨}}) = (−1)^{a}X_{ϵβ^{∨}} .

It follows that

δ_{2}(X_{α^{∨}}) = X_{α^{∨}} .

We take ϵ_{2} = δ_{2}. The lemma also implies that

ϵ_{2}X_{β_{1}^{∨}}ϵ_{2}^{−1} = X_{β_{i}^{∨}}, 2 ≦ i ≦ k .

Set

δ_{3} = ∏_{i=1}^{k} φ_{β_{i}^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned} .

Because ^{L}G^{0} is an adjoint group

λ^{∨}(δ_{3}^{2}) = (−1)^{⟨∑ β_{i},λ^{∨}⟩} = 1, λ^{∨} ∈ X^{∗}(^{L}T^{0}) ,

and δ_{3}^{2} = 1. Let

δ_{3}X_{γ^{∨}}δ_{3}^{−1} = c(γ^{∨})X_{ω_{3}γ^{∨}}, γ^{∨} ∈ 𝔛_{0} .

Certainly

c(ω_{3}γ^{∨}) = c(γ^{∨})^{−1}

and, because a_{k} = 2, we may define t_{3} by

γ^{∨}(t_{3}) = \{ \begin{matrix}1, & γ^{∨} ∈ D, γ^{∨} ∉ 𝔛_{0}, \\ c(γ^{∨})^{−1}, & γ^{∨} ∈ 𝔛_{0} . \end{matrix}

Then we take ϵ_{3} = δ_{3}t_{3}. Finally we take ϵ_{1} = ϵ_{3}ϵ_{2}ϵ_{3}^{−1} = ϵ_{3}ϵ_{2}ϵ_{3}. To show that the cocycle splits it is enough to show that

ϵ_{1}ϵ_{2} = ϵ_{2}ϵ_{1} .

The left side is

ϵ_{3}ϵ_{2}ϵ_{3}ϵ_{2}

and the right

ϵ_{2}ϵ_{3}ϵ_{2}ϵ_{3} .

Now

ϵ_{2}ϵ_{3}ϵ_{2} = φ_{β^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned} ∏_{i=2}^{k} φ_{β_{i}^{∨}} \begin{aligned}( \\ ( & \begin{matrix}0 & 1 \\ −1 & 0 \end{matrix} & ) \\ & & ) \end{aligned} ω_{2}(t_{3}) .

Since any two of the roots β^{∨}, β1^{∨}, …, βl^{∨} are strongly orthogonal we are reduced to verifying the equality

t_{3}ω_{3}ω_{2}(t_{3}) = ω_{2}(t_{3})ω_{2}ω_{3}ω_{2}(t_{3})

or

c(γ^{∨})c(ω_{2}ω_{3}γ^{∨}) = c(ω_{2}γ^{∨})c(ω_{2}ω_{3}ω_{2}γ^{∨})

for all γ^{∨} ∈ 𝔛_{0}. This is clear if γ^{∨} ∈ {α2^{∨}, …, αl−2^{∨}} for then ω_{2} may be removed without affecting either side.

If γ^{∨} ∈ {α^{∨}, α1^{∨}} then

γ^{∨} = ω_{2}γ^{∨}

and the equation is trivially valid. If γ^{∨} ∈ {αl−1^{∨}, αl^{∨}} then

ω_{2}ω_{3}γ^{∨} = ω_{3}γ^{∨}, ω_{2}ω_{3}ω_{3}γ^{∨} = ω_{3}ω_{2}γ^{∨}

and both sides are equal to 1.

If l = 2k + 1 we start with

β^{∨} = x_{1} − x_{l}, β1^{∨} = x_{1} + x_{l}, …, βk^{∨} = x_{k} + x_{k+2} ,

but the argument is otherwise the same.

4) E_{6}.

α^{∨} α2^{∨} α1^{∨}  α3^{∨}  α4^{∨}  α5^{∨}  α6^{∨}^{o}  ^{o}  ^{o}  ^{o}  ^{o}^{o}^{o}

α^{∨} + α1^{∨} + 2α2^{∨} + 2α3^{∨} + 3α4^{∨} + 2α5^{∨} + α6^{∨} = 0

𝔛_{0} must contain α^{∨}, α1^{∨} and α6^{∨}. The group A_{2} is {1} or Z_{2} and the group A_{3} is {1} or Z_{3}. If the group A_{2} is {1, ω} then we may with no loss of generality assume that ω fixes α^{∨} and α2^{∨}. By construction there is an ϵ in ^{L}G′ of order two such that ϵ acts on ^{L}T^{0} as ω and such that

ϵX_{α_{1}^{∨}}ϵ^{−1} = X_{ωα_{1}^{∨}} 1 ≦ i ≦ 6 .

Lemma 3 again implies that

ϵX_{α^{∨}}ϵ^{−1} = X_{α^{∨}} .

The cocycle is therefore trivial on A_{2}.

We now consider A_{3}, which we suppose is Z_{3}. If Z/Z^{0} has order prime to 3 the cocycle is certainly trivial. Thus we may assume that α2^{∨}, α3^{∨}, α5^{∨} belong to 𝔛_{0} but that α4^{∨} does not. We are going to realize A_{3} in the centralizer of α_{4}, regarded as an element of the Lie algebra of ^{L}T^{0}.

Running through the table of positive roots of E_{6} given in [5], we find that the following are orthogonal to α_{4}:

0  0
10000  00001
1  1  1
01100  01110  00110
1  0  0  1
11100  11110  01111  00111

0 11111

1  1
12210  01221
1  1
12211  11221

2 12321

Thus the centralizer is of type A_{5}. With the standard representation of the root system of type A_{5}

1

x_{1} − x_{2} ↔ 01100

0

x_{2} − x_{3} ↔ 10000

0

x_{3} − x_{4} ↔ 01110

0

x_{4} − x_{5} ↔ 00001

1

x_{5} − x_{6} ↔ 00110

Thus a generating element of A_{3} corresponds to the permutation

1 → 2 → 4 → 1, 3 → 5 → 6 → 3 .

This we can realize in SL(6) and therefore certainly in ^{L}G^{0} by an element δ of order 3. As usual we let

δX_{γ^{∨}} = c(γ^{∨})X_{ωγ^{∨}}, γ^{∨} ∈ 𝔛_{0} ,

and define t by

γ^{∨}(t) = \{ \begin{matrix}1, & γ^{∨} ∈ D, γ^{∨} ∉ 𝔛_{0}, \\ c(γ^{∨})^{−1}, & γ^{∨} ∈ 𝔛_{0}. \end{matrix}

Then ϵ = δt has order 3 and can serve as a representative of ω.

5) E_{7}.

o  ^{o}  ^{o}  ^{o}  ^{o}  ^{o}  ^{o}^{o} α2^{∨} α^{∨}  α1^{∨}  α3^{∨}  α4^{∨}  α5^{∨}  α6^{∨}  α7^{∨}

α^{∨} + 2α1^{∨} + 2α2^{∨} + 3α3^{∨} + 4α4^{∨} + 3α5^{∨} + 2α6^{∨} + α7^{∨} = 0

𝔛_{0} must contain α^{∨} and α7^{∨}. Moreover if A is not trivial then it is Z_{2} and [Z : Z^{0}] is prime to 2 unless α3^{∨} and α5^{∨} also belong to 𝔛_{0}. In order to construct a representative of the generator ω of A we work within the centralizer of α_{2} and α_{4}.

We work through the list of positive roots given in [5] and find that those orthogonal to α_{2} and α_{4} are the following:

0  0  0
000001  000010  100000

0 000011

1  1
012210  122100
1  1  1
012211  112210  122110
1  1  1
012221  112211  122111

1 112221 2 134321 2 234321 We obtain a system of type A_{5} and

1

x_{1} − x_{2} ↔ 122100

0

x_{2} − x_{3} ↔ 000010

0

x_{3} − x_{4} ↔ 000001

1

x_{4} − x_{5} ↔ 112210

0

x_{5} − x_{6} ↔ 100000

The image δ of

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

in SL(6) is a representative of ω. δ^{2} is the image of

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

and it is easily checked that this is 1. Since the coefficients a_{2} and a_{4} are even we can define t and ϵ as usual.

The proof of Proposition 1 will be completed by arguments which have nothing to do with semi-simple groups but rely rather on our knowledge of Galois cohomology. We now have a cocycle {a_{σ_{1},σ_{2}}} with values in Z^{0} and we want to show that the inflated cocycle {a_{w_{1},w_{2}}} is trivial.

Lemma 4 Suppose F is a local or a global field and K a finite Galois extension. Let S = Hom(X, C^{×}) be a torus over C. Let Gal(K/F) act on X and hence on S. Let the Weil group W_{K/F} act on S through its projection on Gal(K/F). If {a_{σ_{1},σ_{2}}} is a 2-cocycle of Gal(K/F) with values in S then there is a continuous function b(w) on W_{K/F} with values in S such that

a_{σ_{1},σ_{2}} = b(w_{1})w_{1}(b(w_{2}))b(w_{1}w_{2})^{−1}

for all w_{1}w_{2}.

Variants of this lemma had been drawn to my attention by both Deligne and Hoechsmann, who proved them by means of the dualities of Poitou and Tate. Such methods may well work in general, but it is easier for me to draw on a theorem from [8].

If K is local let C_{K} be the multiplicative group of K and if K is global let it be the group of idèle classes. If H_{c}^{1}(W_{K/F} , S) denotes the group of continuous 1-cocycles of W_{K/F} in S modulo coboundaries then, according to Theorem 1 of [8], there is a canonical isomorphism of H_{c}^{1}(W_{K/F} , S) with the group of characters of the topological group

Hom_{Gal(K/F)}(X^{∨}, C_{K}) .

Here

X^{∨} = Hom(X, Z) .

The characters are not necessarily of absolute value 1. The isomorphism is functorial in S.

If we have an exact sequence

0 → X_{2} → X_{1} → X → 0

in which X_{1}, X_{2} are also Gal(K/F)-modules free over Z, then

1 → Hom_{Gal(K/F)}(X2^{∨}, C_{K}) → Hom_{Gal(K/F)}(X1^{∨}, C_{K}) → Hom_{Gal(K/F)}(X^{∨}, C_{K})

is also exact. Passing to the group of characters we infer from standard facts about extensions of characters that

(1)  H_{c}^{1}(W_{K/F} , S_{1}) → H_{c}^{1}(W_{K/F} , S_{2})

is surjective.

To deduce the lemma from this we need only choose X_{1} correctly. We can clearly choose it to be free over the group ring Z(Gal(K/F)). Then S_{1} is also induced and hence homologically trivial. Consequently in S_{1}

a_{σ_{1},σ_{2}} = c(σ_{1})σ_{1}(c(σ_{2}))c(σ_{1}σ_{2})^{−1} .

If c̄(σ) denotes the image of c(σ) in S_{2} then {c̄(σ)} is a 1-cocycle of Gal(K/F) and thus of W_{K/F}. By the surjectivity of (1) there is a continuous 1-cocycle d(w) of W_{K/F} with values in S_{1} such that

d̄(w) = c̄(σ)ā^{−1}σ(ā)

if w → σ. Here ā is a fixed element of S_{2}. Since S_{1} → S_{2} is surjective, ā is the image of some a ∈ S_{1}. We may replace c(σ) by c(σ)a^{−1}σ(a) and suppose that

d̄(w) = c̄(σ) .

Then w → b(w) = d^{−1}(w)c(σ) is a continuous 1-cochain with values in S whose boundary is the inflation of {a_{σ_{1},σ_{2}}}.

With the proof of Proposition 1 the principal purpose of this note is achieved, but there are some supplementary remarks to be made. First a comment on the role of g_{1}. The influence of g_{1} on groups ^{L}H^{0} and ^{L}H is through ψ′. Suppose we replace g_{1} by ḡ_{1} = wg_{1} and ψ′ by ψ̄′ = ad w ∘ψ′, with w in the normalizer of T^{G_{1}} in G_{1}. If ω is the element of Ω(T_{1}, G_{1}) ≃ Ω(^{L}T^{0},^{L}T^{0},) represented by w then ω_{T/G}(σ) is replaced by

ω̄_{T/G}(σ) = ωω_{T/G}(σ)ω^{−1}

and κ′ by κ̄′ = ω(κ̄′). Let ^{L}H^{0} be replaced by ^{L}H̄^{0}. We write ω = ω_{1}ω_{2} where ω_{2} lies in the Weyl group of LH^{0} and ω_{1} takes positive roots of ^{L}T^{0} in ^{L}H^{0} to positive roots of ^{L}T^{0} in ^{L}H̄^{0}. Then

ω̄_{T/G}(σ) = (ω_{1}ω^{1}(σ)ω_{1}^{−1})(ω_{1}ω^{1}(σ)^{−1}ω_{2}ω^{1}(σ)ω^{2}(σ)ω_{2}^{−1}ω_{1}^{−1}) .

The expression within the second parentheses lies in the Weyl group of ^{L}T^{0} in ^{L}H̄^{0}, and the expression within the first parentheses takes positive roots of ^{L}T^{0} in ^{L}H^{0} to positive roots. Thus

ω̄^{1}(σ) = ω_{1}ω^{1}(σ)ω_{1}^{−1} .

In order to interpret ^{L}H^{0} or ^{L}H̄^{0} as the connected components of associate groups we need to choose in addition root vectors X_{α^{∨}} and X_{}_{ᾱ^{∨}} corresponding to the simple roots. Let u be a representative of ω_{1} in LG^{0} such that

Ad u(X_{α^{∨}}) = X_{}_{ᾱ^{∨}}

if ᾱ^{∨} = ω_{1}α^{∨}. The isomorphism ζ: h → uhu^{−1} of ^{L}H^{0} with ^{L}H̄^{0} may then be extended in a natural fashion to

ζ: ^{L}H → ^{L}H̄ .

We extend the imbedding ξ̄: ^{L}H̄^{0} → ^{L}G^{0} to ξ̄: ^{L}H̄ → Ḡ^{L} by setting

ξ̄(h) = u(ξζ^{−1}(h))u^{−1} .

The conclusion is that g_{1} has no real influence. To each choice Proposition 1 assigns a set of ξ. It is not the individual ξ which matter but only the orbits under conjugation by elements of ^{L}G^{0}, and the preceding discussion yields a canonical bijection between the collections of orbits arising from two different choices.

In order to apply Proposition 1 and the hypothetical principle of functoriality in the associate group effectively, we shall need a way of reducing the study of irreducible representations or of automorphic forms to groups G for which ^{L}G^{0} has a connected center.

We start from a given G and set

P = X_{∗}(T^{G_{1}})/X_{∗}(T_{sc}^{G_{1}}) .

Stable conjugacy—definitions and lemmas 22 The difficulty arises when P is not torsion-free. We represent the Gal(K/F)-module P as a quotient

0 → M′ → Q′ → P → 0

with Q′ torsion-free. We then introduce an imbedding

0 → M′ ⟶^{η} M_{∗}

with M_{∗} induced and M_{∗}/M′ torsion free. Set

Q = M_{∗} ⊕ Q′/{(η(m), m)} .

Q is again torsion free and we clearly have an exact sequence

0 → M_{∗} → Q ⟶^{ϵ_{2}} P → 0 .

If ϵ_{1} is the homomorphism X_{∗}(T^{G_{1}}) → P set

X̃_{∗} = {(λ, μ) ∈ X_{∗}(T^{G_{1}}) ⊕ Q|ϵ_{1}(λ) = ϵ_{2}(μ)}

and set

X̃_{∗} = Hom(X̃_{∗}, Z) .

We certainly have

X_{∗}(T_{sc}^{G}) ↪ X̃_{∗}

by means of the map to the first factor as well as a surjection

X̃_{∗} → X_{∗} → 0

with kernel M_{∗}. Dual to this we have

0 → X^{∗} → X̃^{∗} → M^{∗} → 0 .

Here

M^{∗} = Hom(M_{∗}, Z) .

There is clearly a central extension G̃_{1} of G_{1} over F such that if T̃^{G_{1}} is the inverse image of T^{G_{1}} then X^{∗}(T̃^{G_{1}}) = X̃^{∗}

and the contragredient to T̃^{G_{1}} → G^{G_{1}} is X^{∗}(T^{G_{1}}) = X^{∗} → X̃^{∗} = X^{∗}(T̃^{G_{1}}) .

We define G̃ over F by twisting G̃_{1} by the cocycle {ψ ∘ (ψ^{−1})} in the adjoint group of G_{1} and obtain a commutative diagram

G̃ ⟶^{ψ̃} G̃_{1}
↓_{φ} ↓^{φ_{1}}
G ⟶_{ψ} G_{1}

with vertical arrows defined over F. If Z is the kernel of φ or of φ_{1}, for they are isomorphic, then

M^{∗} = X^{∗}(Z) .

Since M^{∗} is induced we infer from the Tate-Nakayama theory and Hilbert’s Theorem 90 that if F is a local or a global field then

G̃(F) → G(F)

is surjective and that if it is global then

G̃(A_{F}) → G(A_{F})

is surjective as well. This allows us when F is local to identify representations of G(F) as representations of G̃(F) which are trivial on Z(F), and when F is global to identify automorphic representations of G(A_{F}) with automorphic representations of G̃(A_{F}) trivial on Z(A_{F}).

If T is any Cartan subgroup of G with inverse image T̃ then

1 → Z → T̃ → T → 1

is exact and

H^{1}(Gal(F̄/F), T̃(F̄)) → H^{1}(Gal(F̄/F), T (F̄))

is injective. Consequently

𝔇(T̃) = 𝔇(T)

and

𝔈(T̃) = 𝔈(T) .

There is another way of expressing the last relation.

Certainly X_{∗}(T̃_{sc}) = X_{∗}(T_{sc}). I claim that the groups

{λ ∈ X_{∗}(T̃_{sc})|λ = ∑_{Gal(K/F)} ω_{T̃/G}(σ)μ̃(σ) − μ̃(σ), μ̃(σ) ∈ X_{∗}(T̃)}

and

{λ ∈ X_{∗}(T_{sc})|λ = ∑_{Gal(K/G)} ω_{T/G}(σ)μ(σ) − μ(σ), μ(σ) ∈ X_{∗}(T)}

are also equal. Since

X_{∗}(T̃) = {(λ, μ) ∈ X_{∗}(T) ⊕ Q|ϵ_{1}(λ) = ϵ_{2}(μ)} ,

the first group is certainly contained in the second. On the other hand, if μ(σ) ∈ X_{∗}(T) and

λ = ∑ ω_{T/G}(σ)μ(σ) − μ(σ) ∈ X_{∗}(T_{sc}) ,

then ϵ_{1}μ: σ → ϵ_{1}(μ(σ)) defines an element of H^{−2}(Gal(K/F), P). Since

H^{−2}(Gal(K/F), Q) → H^{−2}(Gal(K/F), P)

is surjective, there exists ν(σ) in Q such that

∑ σν(σ) − ν(σ) = 0

and

ϵ_{2}(μ) − ϵ_{1}(ν) = δη

is a boundary. Here η: (σ_{1}, σ_{2}) → η(σ_{1}, σ_{2}) is a 2-chain. If η′ is a 2-chain with values in X_{∗}(T) with ϵ_{1}(η)′ = η, we may replace μ by μ − δη′ without affecting λ and hence assume that

ϵ_{2}(μ) = ϵ_{1}(ν) .

Then

μ̃ = μ ⊕ ν

defines a chain of

X_{∗}(T̃) ⊆ X_{∗}(T) + Q

and

∑ ω_{T̃/G}(σ)μ̃(σ) − μ̃(σ) = λ .

Finally, I add a few remarks that it is useful to bear in mind when applying the constructions of this paper to groups in whose definition a restriction of scalars intervenes. Suppose F is a finite extension of E and Ḡ is the group over E obtained from G by restriction of scalars. Then

T Ḡ =_{F/E} T^{G}

is a Cartan subgroup of Ḡ over E. Moreover, Ḡ_{1} =_{F/E} G_{1} is quasi-split, and T Ḡ_{1} =_{F/E} T^{G_{1}} is a Cartan subgroup of it.

Once we have fixed an imbedding of F in Ē we may identify Ḡ(Ē) with the set of functions φ from Gal(Ē/E) to G(Ē) = G(F̄) satisfying

φ(στ) = σ(φ(τ)) σ ∈ Gal(Ē/F) .

Ḡ_{1}(Ē) is obtained in a similar fashion. Recall that in Lemma 2.3 of [10] we have associated to ψ: G → G_{1} an isomorphism ψ̄: Ḡ_{1} over Ē. If I is a set of representatives for the cosets of Gal(Ē/F) in Gal(Ē/E), then ψ̄ takes φ to φ_{1} with

φ_{1}(τ) = ψ(φ(τ)), τ ∈ I .

If ḡ_{1} is the function in Ḡ_{1}(Ē) which takes τ ∈ I to g_{1}, then

ψ̄′ = ad ḡ_{1} ∘ ψ̄

is obtained from ψ′ in just the same way that ψ̄ is obtained from ψ. It does depend on the choice of coset representatives, but that is not important. Let us fix I for now.

It was observed in [10] that, as Gal(Ē/E)-modules,

X_{∗}(T Ḡ) = (Gal(Ē/E), Gal(Ē/F), X_{∗}(T^{G}))
X_{∗}(T Ḡ_{1}) = (Gal(Ē/E), Gal(Ē/F), X_{∗}(T^{G_{1}})) .

Both these modules consist of functions on Gal(Ē/E), and if ψ̄′ takes λ to λ_{1}, then

λ_{1}(τ) = ψ′(λ(τ)), τ ∈ I .

Shapiro’s Lemma shows that

λ → ∑_{τ∈I} λ(τ)

yields an isomorphism 𝔈(T Ḡ)⟶^{∼}𝔈(T^{G}). Thus κ pulls back to

κ̄: λ → ∑_{τ∈I} κ(λ(τ))

and κ′ to

κ̄′: λ → ∑_{τ∈I} κ′(λ(τ)) .

LḠ^{0} consists of the functions φ on Gal(Ē/E) with values in ^{L}G^{0} satisfying

φ(στ) = σ(φ(τ)), σ ∈ Gal(Ē/F) .

It is clear that ^{L}H̄^{0} consists of those φ for which

(2)  φ(τ) ∈ ^{L}H^{0}

for all τ ∈ I.

If

τσ = α_{τ} (σ)τ′

with α_{τ} (σ) ∈ Gal(Ē/F) and τ′ ∈ I and T̄ = T Ḡ, then ω_{T̄/Ḡ}(σ) takes λ to λ′ with

λ′(τ) = ω_{T/G}(α_{τ} (σ))λ(τ′), τ ∈ I .

If

ω_{T̄/Ḡ}(σ) = ω̄^{1}(σ)ω̄^{2}(σ) ,

where ω̄^{2}(σ) lies in the Weyl group of ^{L}T̄^{0} in ^{L}H̄^{0} and ω̄^{1}(σ) takes positive roots in ^{L}H̄^{0} to positive roots, then ω̄^{1}(σ) takes λ to λ′ with

λ′(τ) = ω^{1}(α_{τ} (σ))λ(τ′), τ ∈ I .

Thus ^{L}H̄ is the associate group attached to ^{L}H by the functor 𝔊^{∨}(F) → 𝔊^{∨}(E) of [10]. Consulting the definitions of [9], we see that if ξ: ^{L}H → ^{L}G extends ^{L}H^{0} ↪ ^{L}G^{0} then the homomorphism ξ̄: ^{L}H → ^{L}G associated to ξ by the functorial process of [10] extends ^{L}H^{0} ↪ ^{L}G^{0}.

The conclusion is that the constructions of this paper behave simply under restriction of scalars, as one expects. It should also be noticed that the functorial constructions of [10] also allow one to construct the homomorphism of Proposition 1 even in situations that do not strictly arise from restriction of scalars. They can sometimes be used for connected subgroups of Ḡ =_{F/E} G with abelian quotients. We will meet an example of this in another paper.

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[2] N. Bourbaki, Groupes et algèbres de Lie, chap. 4, 5, et 6, Hermann (Paris).

[3] E. Dynkin, Semi-simple subalgebras of semi-simple Lie algebras, A.M.S. Trans. Ser. 2, 6 (157).

[4] D. Flath, Thesis, Harvard (1977).

[5] H. Freudenthal and H. de Vries, Linear Lie groups, Academic Press (New York).

[6] H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Springer Lecture Notes 114 (1970).

[7] J.-P. Labesse and R. P. Langlands, L-indistinguishability for SL(2), to appear in Can. J. Math.

[8] R. P. Langlands, Representations of abelian algebraic groups, preprint (1968).

[9] , Problems in the theory of automorphic forms, Springer Lecture Notes 170.

[10] , On the classification of irreducible representations of real algebraic groups, preprint (1973).

[11] R. Steinberg, Regular elements of semi-simple algebraic groups, Publ. Math. I.H.E.S. (1965).

[12] D. Shelstad, Characters and inner forms of a quasi-split group over R, to appear in Comp. Math.

[13] , Orbital integrals and a family of groups attached to a real reductive group, to appear.

[14] , Notes on L-indistinguishability (based on a lecture of R. P. Langlands), to appear in Proc. of A.M.S. Summer Institute (Corvallis).