L-Indistinguishability for SL(2)*

J.-P. Labesse and R.P. Langlands

1. Introduction. The notion of L-indistinguishability, like many others current in the study of L-functions, has yet to be completely defined, but it is in our opinion important for the study of automorphic forms and of representations of algebraic groups. In this paper we study it for the simplest class of groups, basically forms of SL(2). Although the definition we use is applicable to very few groups, there is every reason to believe that the results will have general analogues [12].

The phenomena which the notion is intended to express have been met — and exploited — by others (Hecke [5] §13, Shimura [17]). Their source seems to lie in the distinction between conjugacy and stable conjugacy. If F is a field, G a reductive algebraic group over F, and F̄ the algebraic closure of F, then two elements of G(F) may be conjugate in G(F̄) without being conjugate in G(F). In addition, if F is a local field then in many cases there is a rough duality between conjugacy classes in G(F) and equivalence classes of irreducible representations of G(F), and one might expect the coarse classification of stable conjugacy to lead to a grouping of these equivalence classes. One of the groups is now called an L-packet and the elements in it are said to be L-indistinguishable because in the cases that are understood they have the same L-functions.

It was the L-packets with which we started. If G is GL(2), or even GL(n), then stable conjugacy is the same as conjugacy and the L-packets will consist of a single element, and there is no need to introduce them. They do not appear in [6]. The group GL(2, F) acts on SL(2, F) by g : h → h^{g} = g^{−1}hg and, if F is a local field, on the irreducible representations of SL(2, F) by π → π^{g} with π^{g}(h^{g}) = π(h). Two elements of SL(2) are stably conjugate if and only if they lie in the same orbit under GL(2, F) and it is expedient to define two irreducible representations of SL(2, F) to be L-indistinguishable if they lie in the same orbit under GL(2, F), or more precisely, if the induced representations of the Hecke algebra lie in the same orbit. This definition can only be provisional but it will serve our purpose, which is to explore the notion for SL(2) and some related groups thoroughly, attempting to formulate and verify theorems which are likely to be of general validity.

Our original purpose was more specific. Suppose F is a global field and π = ⊗π_{v}, the product being taken over all places of F, is an automorphic representation of SL(2, A_{F}). If for each v we choose a π′_{v} which is L-indistinguishable from π_{v} and equivalent to it for almost all v then π′ = ⊗π′_{v}

L-indistinguishability 2

* Appeared in Can. J. Math., vol XXXI(1979). Included by permission of Canadian Mathematical Society. might or might not be an automorphic representation. We wished to show that it is, except for a very special class of π, those associated to characters of the group of idèles of norm one in a quadratic extension, and this we could do without too much difficulty. The problem was posed and solved in the spring of 1971 while we were together at the Mathematical Institute in Bonn in the Sonderforschungsbereich Theoretische Mathematik, and the paper could have been written then, except that we could not formulate the results in a satisfying fashion. For this the groups H of [12] are needed, for which an adequate general definition was found only after many conversations with Shelstad, as well as the groups S and S^{0}, whose introduction was suggested by the work of Knapp-Zuckerman [7].

Because we had some specific applications in mind we have considered groups slightly more general than twisted forms of SL(2), but they can be left to the body of the paper. If G is SL(2) or a twisted form then the L-group ^{L}G can, for the present purpose, be taken to be P GL(2, C). If F is a local field and φ a homomorphism of the Weil group W_{F} into ^{L}G there is in general (cf. [18]) an associated L-packet Π(φ), and according to the results of §3 it will contain only finitely many equivalence classes. If S_{φ} is the centralizer of φ(W_{F}) in ^{L}G and S_{φ}^{0} the connected component of the identity in S_{φ} then, as will be seen in §6 and §7, there is a pairing ⟨s, π⟩ between S_{φ}^{0}\S_{φ} and Π(φ) which is often but not always a duality.

This local pairing is of interest in itself, and is also of some significance in global multiplicity questions. To form a global L-packet Π one chooses local L-packets Π_{v}, such that Π_{v} contains the unramified representation π_{v}^{0} for almost all v, and takes Π to be the collection

Π = {π = ⊗π_{v} | π_{v} ∈ Π_{v} for all v and π_{v} = π_{v}^{0} for almost all v}.

Some of the π may be automorphic and others not. It is shown in §6 and §7 that they are automorphic simultaneously unless Π is the L-packet Π(φ) associated to a homomorphism φ : W_{F} → ^{L}G = P GL(2, C) obtained from an irreducible induced two-dimensional representation of W_{F}. If φ_{v} is the restriction of φ to the decomposition group at v and π = ⊗π_{v} lies in Π(φ) then π_{v} lies in Π(φ_{v}) and S_{φ} ⊆ Sφ_{v}, S_{φ}^{0} ⊆ S^{0}φ_{v}. We may define ⟨s, π⟩ to be Π_{v}⟨s, π_{v}⟩. One of the principal conclusions of this paper is that the multiplicity with which π occurs in the space of cusp forms is

\frac{1}{[S_{φ} : S_{φ}^{0}]} ∑_{s∈S_{φ}^{0}\S_{φ}}⟨s, π⟩.

One hopes that a similar result is valid for every L-packet containing an automorphic representation, but even its formulation would demand the introduction of the problematical group G_{Π(F)} of §2 of [15].

Although the main results of the paper are in §6 and §7, the technical burden is carried by §5, in which the analysis of the trace formula suggested in [12] is carried out at length. The trace formula seldom functions without some local harmonic analysis, but usually with much less than appears necessary at first sight, and once it is primed it will start to pump out many local results. Since we are dealing with an easy group for which we could establish many of the local results directly, we have done so. For other groups, where local information is harder to come by, it will be necessary to bring the trace formula into play sooner, and so the reader who has his eye on generalizations should not spend too much time on the details of §2, §3, and §4. The critical observations are that the function Φ^{T′}(f) : γ → Φ^{T′}(γ, f) is smooth and that the map on distributions dual to f → Φ^{T′}(f) sends a character to a difference of characters.

Finally we observe that [8] and [16] serve to some extent as introductions to this paper and that to avoid technical complications we have confined ourselves to fields of characteristic zero.

2. Local theory. Let F be a local field of characteristic zero and G the group SL(2). Let T be a Cartan subgroup of G defined over F. Since G is simply-connected and

H^{1}(F, G) = {1}

the two sets 𝔇(T) and 𝔈(T) introduced in [12] are equal to each other and to

H^{1}(F, T).

Let G̃ be the group GL(2). Then the centralizer T̃ of T in G̃ is a Cartan subgroup of G̃. Since

H^{1}(F, T̃) = 1

any g in 𝔄(T) ([12]) may be written as a product sh with s ∈ T̃(F̄) and h in G̃(F). Conversely any h in G̃(F) is a product s^{−1}g with g ∈ G(F̄) and s ∈ T̃(F̄). The element g must lie in 𝔄(T) for

h^{−1}th = g^{−1}tg t ∈ T (F̄).

If L is the centralizer of T (F) in the algebra of 2 × 2 matrices over F then

{det t|t ∈ T̃(F)} = {Nm_{L/F} x|x ∈ L^{×}}

and

g → det h (modNm_{L/F} L^{×})

yields an isomorphism

𝔇(T) = 𝔈(T) ≃ F^{×}/Nm_{L/F} L^{×}.

More generally we could suppose that F was an extension of some field E and then consider a group G′ over E with

Res_{L/E}G ⊆ G′ ⊆ Res_{L/E}G̃.

Thus G′ is defined by a subgroup A of Res_{L/E}G_{m} and

G′(F) = {g ∈ G̃(F)|det g ∈ A(E)}.

If T′ is the centralizer of Res_{F/E} T in G′ then one shows, just as above, that

𝔇(T′) = 𝔇(T′/E) ≃ F^{×}/A(E)Nm_{L/F} L^{×}.

For our purposes it is best simply to take a closed subgroup A of F^{×} and to let

G′ = {g ∈ G̃(F)|det g ∈ A},

so that G′ may no longer be the set of points on an algebraic group rational over some field. T′ will be the intersection of G′ with T̃(F) and we set

𝔇(T′) = T̃(F)\G̃(F)/G′ ≃ F^{×}/A Nm_{L/F} L^{×}.

It is a group, and is either trivial or of order two.

We return for a moment to G. Suppose, as in [12], that κ is a homomorphism of X_{∗}(T) into C^{×} that is invariant under the Galois group. There are two possibilities.

a) T is split and the Galois group acts trivially. Then κ is any homomorphism of X_{∗}(T) into C^{×}. On the other hand X_{∗}(T) has no elements of norm 0, 𝔇(T) is trivial, and so κ restricted to 𝔇(T) is also trivial.

b) T is not split. Then the action of the Galois group factors through 𝔊(L/F) = {1, σ} and σ acts as −1. Thus κ is of order 2, every element is of norm 0, and

𝔇(T) = X_{∗}(T)/2X_{∗}(T)

is of order 2. Since neither root α^{∨} lies in 2X_{∗}(T), κ(α^{∨}) ≠ −1 if and only if κ is not trivial.

The group H associated to the pair T, κ ([12]) is either G or T, and we shall only be interested in the case that it is T. Yet in the following discussion it is the restriction of κ to 𝔇(T) which plays a role, and this is not enough to determine H. What we do is introduce a character κ′ of 𝔇(T′) and assume that if T is not split then κ′ is not trivial.

We fix Haar measures on G′ and T′ and let γ be a regular element in T′. If h ∈ G̃(F) we may transfer the measure from T′ to h^{−1}T′h. If f is a smooth function on G′ with compact support and δ is the image of h in 𝔇(T′) we set

Φ^{δ}(γ, f) = ∫_{h^{−1}T′h\G′} f(g^{−1}h^{−1}γhg)dg.

We are going to introduce a function d(γ) on the set of regular elements of T′ and will set

Φ^{T′κ′}(γ, f) = Φ^{T′}(γ, f) = d(γ) ∑_{𝔇(T′)} κ′(δ)Φ^{δ}(γ, f).

Let γ_{1} and γ_{2} be the eigenvalues of γ. If T is split

d(γ) = |(γ_{1} − γ_{2})^{2}|^{1/2}/|γ_{1}γ_{2}|^{1/2}.

If d is not split the definition is more complicated, and requires several choices to be made. κ′ may now be regarded as the non-trivial character of F^{×}/Nm L^{×}. Let γ^{0} be a fixed regular element in T̃(F) and let ψ be a fixed non-trivial additive character of F. The factor

λ(L/F, ψ)

has been introduced in [13]. Moreover, an order on the eigenvalues γ_{1}^{0}, γ_{2}^{0} of γ^{0} determines an order γ_{1}, γ_{2} on those of γ. Set

d(γ) = λ(L/F, ψ)κ′ (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) \frac{|(γ_{1} − γ_{2})^{2}|^{1/2}}{|γ_{1}γ_{2}|^{1/2}} .

Different choices of ψ and γ^{0} lead either to d(γ) once again or to −d(γ). The change of sign is not important.

Lemma 2.1 We may extend

γ → Φ^{T′}(γ, f)

to a smooth function on T′ with compact support.

What we must do is define Φ^{T′}(γ, f) when γ is a scalar matrix in G′ and show that the resultant function is smooth in the neighbourhood of such a γ. This is not difficult but some care must be taken with the normalization of measures. The coset space T′\G′ is open in T̃(F)\G̃(F) and the given measure on T′\G′ defines one on T̃(F)\G̃(F). We may write

Φ^{T′}(γ, f) = d(γ) ∫_{_{T}(F)/_{G}(F)} f(g^{−1}γg)κ′(det g)dg.

It is enough to prove the lemma for one choice of the measure on T̃(F)\G̃(F). For a given regular γ, T̃(F)\G̃(F) may be identified with the orbit O(γ) of γ on G̃(F) under conjugacy. We may assume that the measure on T̃(F)\G̃(F) is

|ω_{γ}|/|γ_{1} − γ_{2}|

if ω is defined as in Lemma 6.1 of [9] and ω_{γ} as on p. 77 of the same paper. Then

Φ^{T′}(γ, f) = λ(L/F, ψ)κ′ (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) \frac{1}{|γ_{1}γ_{2}|^{1/2}} ∫_{O(γ)} ϵ(h)f(h)|ω_{γ}|

if

ϵ(g^{−1}γg) = κ′(det g).

It is understood that

λ(L/F, ψ)κ′ (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) = 1

if T̃ is split.

If a ∈ F^{×} let

γ(a) = a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) .

The form ω_{γ(a)} is still defined on O(γ(a)).

Definition 2.2. If T̃ is split and a lies in the centre of T′ set

Φ^{T′}(a, f) = \frac{1}{|a|} ∫_{O(γ(a))} f(h)dh.

If T′ is defined by a quadratic extension L, we may regard G̃(F) as the group of invertible linear transformations of L. Then T̃(F) = L^{×}, the elements acting by multiplication. Choose a basis {1, τ} for L over F and let

τ^{2} = uτ + v.

Let γ = a + bτ lie in T̃(F) or L^{×}. Its eigenvalues are then γ_{1} = a + bτ, γ_{2} = a + bτ̄, and

γ_{1} − γ_{2} = b(τ − τ̄).

Moreover, γ corresponds to the matrix

( \begin{matrix}a & bv \\ b & a + bu \end{matrix} ) .

If

g = ( \begin{matrix}a_{1} & b_{1} \\ c_{1} & d_{1} \end{matrix} )

then

h = g^{−1}γg = ( \begin{matrix}∗ & −bNm_{L/F} (b_{1} + d_{1}v)/det g \\ b Nm_{L/F} (a_{1} + c_{1}τ)/det g & ∗ \end{matrix} ) .

If

h = ( \begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix} )

then

κ′ (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) = κ′ (\frac{τ − τ̄}{γ_{1}^{0} − γ_{2}^{0}} ) κ′(c_{2}) = κ′ (\frac{τ − τ̄}{γ_{1}^{0} − γ_{2}^{0}} ) κ′(−b_{2}).

However, an element on O(γ(a)) has the form

g^{−1}γ(a)g = a ( \begin{matrix}∗ & \frac{d_{1}^{2}}{det g} \\ \frac{−c_{1}^{2}}{det g} & ∗ \end{matrix} ) = ( \begin{matrix}a_{2} & b_{2} \\ c_{2} & d_{2} \end{matrix} ) .

If both b_{2} and c_{2} are not zero then their quotient is a square. Moreover, one of them is always different from zero; so

ϵ(h) = κ′(c_{2}) = κ′(−b_{2})

is a well-defined function on O(γ(a)).

Definition 2.3. If T̃ is not split and a lies in the centre of T′ set

Φ^{T′}(a, f) = λ(L/F, ψ)κ′ (\frac{τ − τ̄}{γ_{1}^{0} − γ_{2}^{0}} ) \frac{1}{|a|} ∫_{O(γ(a))} ϵ(h)f(h)|ω_{γ(a)}|.

We have not mentioned it before but we identify G̃(F) with a set of linear transformations of L by choosing a vector x in F^{2} and identifying L with F^{2} by means of γ → γx. The standard basis of F^{2} then yields a basis of L. It is understood in the above discussion that {1, τ} can be obtained from this basis by an element of G′.

With these definitions the function Φ^{T′}(γ, f) is certainly smooth when the support of f does not meet the set of scalar matrices. To prove it in general we have only to show that there exists a function c(a) on F^{×} such that Φ^{T′}(·, f) extends to a smooth function on T′ which equals

c(a) ∫_{O(γ(a))} ϵ(h)f(h)|ω_{γ(a)}|

on F^{×}. Here ϵ(h) is to be identically 1 if T̃ is split.

For split T̃ this is a well-known and basic fact about orbital integrals. If F is R but T̃ is not split it follows readily from Harish-Chandra’s study of orbital integrals for real groups (cf. [4]). For non-archimedean fields and non-split T̃ we carry out the necessary calculation.

Again we regard G̃(F) as the group of invertible linear transformations of L. At a cost of no more than a change of sign for Φ^{T′}(·, f) we may suppose that {1, τ} is a basis over O_{F} of the ring of integers O_{L} in L. Let G̃(O_{F}) be the stabilizer of O_{L} in G̃(F). Replacing f by

g → ∫_{G(O_{F})} f(k^{−1}gk)κ′(det k)dk

if necessary, we may assume that

f(k^{−1}gk) = κ′(det k)f(g), k ∈ G̃(O_{F}).

The calculation now proceeds along the lines of the proof of Lemma 7.3.2 of [6]. If ϖ is a generator of the maximal ideal of O_{F} then every double coset in T̃(F)\G̃(F)/G̃(O_{F}) contains a g such that

gO_{L} = O_{F} + ϖ^{m}O_{F} τ m ≧ 0.

In other words it contains a representative

( \begin{matrix}1 & 0 \\ 0 & ϖ^{m} \end{matrix} ) , m ≧ 0.

It is clear that m is uniquely determined.

If m is unramified the index

δ_{m} = [T̃(F) ( \begin{matrix}1 & 0 \\ 0 & ϖ^{m} \end{matrix} ) G̃(O_{F}) : F^{×}G̃(O_{F})]

is given by

\begin{aligned}δ_{0} & = 1, \\ δ_{m} & = (q + 1)q^{m−1}, m > 0. \end{aligned}

If L is ramified

δ_{m} = 2q^{m}.

Here q is the number of elements in the residue field. Moreover, apart from a constant that does not depend on f or on γ, the function Φ^{T′}(γ, f) is given by

(2.1)  ∑_{m=0}^{∞} κ′(bϖ^{−m})|b|δ_{m}f (( \begin{matrix}a & bvϖ^{m} \\ bϖ^{−m} & a + bu \end{matrix} ))

if γ = a + bτ.

If L is unramified then apart from a factor 2 the index δ_{m} is |ϖ^{−m}|. If |b| = |ϖ|^{N} the above sum is twice

∫_{|x|≧|ϖ|^{N}} κ′(x)f (( \begin{matrix}a & b^{2}v/x \\ x & a + bu \end{matrix} )) dx.

If γ is close to a scalar a_{0} and N therefore very large then

|b^{2}v/x| ≦ ϖ|^{N}|v|

and

f (( \begin{matrix}a & b^{2}v/x \\ x & a + bu \end{matrix} )) = f (( \begin{matrix}a_{0} & 0 \\ x & a_{0} \end{matrix} )) .

Since

(2.2)  ∫_{|x|<|ϖ|^{N}} κ′(x)dx = 0,

the above integral equals

∫ κ′(x)f (( \begin{matrix}a_{0} & 0 \\ x & a_{0} \end{matrix} )) dx.

The desired assertion now follows from simple, standard integration formulae.

If L is unramified the sum (2.1) is equal to

κ′(b)|b|f (( \begin{matrix}a & bv \\ b & a + bu \end{matrix} )) + (1 + \frac{1}{q}) × ∑_{m=1}^{∞} κ′(bϖ^{−m})|bϖ^{−m}|f (( \begin{matrix}a & bϖ^{m}v \\ bϖ^{−m} & a + bu \end{matrix} )) .

The second term is equal to

(1 + \frac{1}{q}) ∫_{|x|≥|ϖ^{−1}|} κ′(x)f (( \begin{matrix}a & b^{2}v/x \\ x & a + bu \end{matrix} )) dx.

We argue as before except that (2.2) is replaced by

∫_{|x|<|bϖ^{−1}|} κ′(x)dx = ∑_{m=0}^{∞} κ′(bϖ^{m})|bϖ^{m}| = κ′(b)|b|/(1 + q^{−1}).

There is a supplement to the lemma which we will have to take account. Suppose f is the restriction to G′ of the characteristic function of G̃(O_{F}) divided by its measure. It is clear that

ϕ^{T′}(γ, f) = 0

if L is ramified. If L is unramified and T̃(F) intersected with G̃(O_{F}), which we denote T̃(O_{F}), corresponds to the units of O_{L} then ϕ^{T′}(γ, f) is 0 unless γ is a unit, but then, if |b| = |ϖ|^{n}, it is given by

(meas T̃(O_{F}))^{−1}{(−1)^{n}q^{−n} + (1 + q^{−1} ∑_{m=1}^{n}(−1)^{n−m}q^{m−n}} = (meas T̃(O_{F}))^{−1}.

The map

f → ϕ^{T′}(·, f) = ϕ^{T′}(f)

induces an adjoint map on distributions. We want to examine its effect on characters. If T̃ is split then every character θ′ of T′ defines a principal series representation π(θ′) of G′ and it is easily seen that

θ′ → χ_{π(θ′)}

if χ_{π(θ′)} is the character of π(θ′). Before considering T̃ that are not split we state and prove some simple lemmas.

Lemma 2.4. If π̃ is an irreducible admissible representation of G̃(F) then the restriction of π̃ to G′ is the direct sum of finitely many irreducible representations.

Lemma 2.5. If π′ is an irreducible admissible representation of G′ then there exists an irreducible admissible representation π̃ of G̃(F) which contains π′.

More general forms of the first lemma are known, but we are dealing with a very simple situation. We may replace G′ by F^{×}G′ and hence suppose that

G̃(F)/G′ ≃ Z_{2}^{n}

with some integer n, for we have assumed that F is of characteristic 0. If the obvious induction is applied, it is enough to prove the two lemmas with G̃(F) replaced by G′′, where G′ ⊆ G′′ ⊆ G̃(F) and

G′′/G′ ≃ Z_{2}.

π̃ is replaced by π′′. Suppose π′′ acts on V. The restriction of π′′ to G′ is admissible. If it is irreducible the first lemma is valid for π′′. Otherwise V contains a non-trivial invariant subspace W. If g ∈ G′′ − G′ then

V = W + π′′(g)W

and

W ∩ π′′(g)W

is invariant under G′′. Thus it must be 0 and

V = W ⊕ π′′(g)W.

It is clear that G′ must act irreducibly on the two summands.

To prove the second lemma we start from π′ and consider

σ = Ind(G′′, G′, π′).

If g ∈ G′′ − G′ and

h → π′(g^{−1}hg), h ∈ G′,

is not equivalent to π′ then σ is irreducible and contains π′. Otherwise if π′ acts on W then

π′(g^{−1}hg) = A^{−1}π′(h)A, h ∈ G′.

We may assume that

A^{2} = π′(g^{2})

and then extend π′ to a representation π′′ of G′′ on W satisfying

π′′(g) = A.

The map from W to the space of σ which takes w to

g → π′′(g)w

yields π′′ as a component of σ. Indeed if ω is the non-trivial character of G′′/G′ it is clear that

σ ≃ π′′ ⊕ (π′′ ⊗ ω).

Lemma 2.6. The restriction of π̃ to G′ contains no representation π′ with multiplicity greater than one.

This lemma is known for archimedean fields. We verify it only for non-archimedean. We may certainly suppose that π̃ is not one-dimensional, and hence that it is infinite-dimensional and possesses a Whittaker model [6].

We also assume once again that G′ ⊇ F^{×}. We start from a given infinite-dimensional π′ and consider

σ = Ind(G̃(F), G′, π′).

By the Frobenius reciprocity law, π′ is contained in a representation π̃ with the same multiplicity that π̃ is contained in σ.

Suppose ψ is a given non-trivial additive character of F and thus of

N(F) = \{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) |x ∈ F \} .

To say that π̃ has a Whittaker model is to say that it is contained in

Ind(G̃(F), N(F), ψ).

However

N(F) ⊆ G′ ⊆ G̃(F)

and every coset of G′ in G̃(F) is represented by a matrix of the form

( \begin{matrix}a & 0 \\ 0 & β \end{matrix} ) .

Since these matrices normalize N(F) we infer from Lemma 2.5 that for some non-trivial character ψ′ of F the representation π′ is a constituent of

Ind(G′, N(F), ψ′).

The transitivity of induction and the uniqueness of the Whittaker model imply that π̃ is contained at most once in σ.

We associate to π′ the group G(π′) of all g ∈ G̃(F) for which

h → π′(g^{−1}hg) h ∈ G′

is equivalent to π′ and we associate to π̃ the set X(π̃) of all characters ω of G̃(F)/G′ for which

π̃ ⊗ ω ≃ π̃.

Any ω in X(π̃) is trivial on squares and hence of order two.

Corollary 2.7. Suppose π′ is a component of Ind(G′, N(F), ψ′). Then it is also a component of Ind(G′, N(F), ψ′_{1}) if and only if ψ′_{1}(x) ≡ ψ′(βx) for some β in {det g|g ∈ G(π′)}.

If ψ′_{1}(x) = ψ′(βx) and π′ is realized on a space V of functions φ satisfying

φ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) = ψ′(x)φ(g)

and if

h = ( \begin{matrix}β & 0 \\ 0 & 1 \end{matrix} )

lies in G(π′) then it is also realized on

{φ′|φ′(g) = φ(hgh^{−1}), φ ∈ V }

and

(2.3)  φ′ (( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) g) = ψ′_{1}(x)φ′(g).

Conversely suppose π′ is realized on V and on a space of functions satisfying (2.3). Then π′ and

π′_{1} : g → π′(h^{−1}gh)

are both contained in

Ind(G′, N(F), ψ′).

The uniqueness of the Whittaker model for G̃(F) implies that π′ ∼ π′_{1} and that h ∈ G(π′).

Lemma 2.8. Suppose π′ is a component of π̃. The character ω belongs to X(π̃) if and only if it is trivial on G(π′). Moreover, the number of components of the restriction of π̃ to G′ is |X(π̃)|.

Let the restriction of π̃ to G′ be a direct sum

π′_{1} ⊕ ⋯ ⊕ π′_{r}

of irreducible representations with π′_{1} = π′. If π̃ acts on V and

V = V_{1} ⊕ ⋯ ⊕ V_{r}

then V_{1} is invariant under G(π′). If π_{1} is the representation of G(π′) on V_{1} then

π̃ = Ind(G̃(F), G(π′), π_{1}).

Consequently

r = [G̃(F) : G(π′)]

and every character of G̃(F)/G(π′) belongs to X(π̃). If ω belongs to X(π̃) and A intertwines π̃ and π̃ ⊗ ω then A : V_{i} → V_{i} and it acts as a scalar on V_{i}. Therefore π_{1} and π_{1} ⊗ ω are not merely equivalent but in fact equal. Hence ω is trivial on G(π′).

Suppose π̃ and σ̃ are two irreducible representations of G̃(F) whose restrictions to G′ contain π′. We may also decompose the restriction of σ̃ to G′ into a direct sum

σ′_{1} ⊕ ⋯ ⊕ σ′_{r}

with r = [G̃(F) : G(π′)] and with σ′_{1} = π′. If σ̃ acts on W and σ′_{1} on W_{1} then W_{1} is invariant under G(π′) and if σ_{1} is the representation of G(π′) on W_{1} then

σ_{1} = ω_{1} ⊗ π_{1}

where ω_{1} is a character of G′\G(π′). Thus

σ̃ = Ind(G̃(F), G(π′), σ_{1})

is equivalent to ω ⊗ π̃ where ω is any extension of ω_{1} from G(π′) to G̃(F). In particular if π̃ and ω ⊗ π̃ have the same restriction to G′ then ω = ω_{1}ω_{2} with ω_{1} trivial on A and ω_{2} ⊗ π̃ ≃ π̃.

Suppose ω is a character of F^{×} of order two and L the corresponding quadratic extension. One knows [6] that to each character θ of L^{×}, of absolute value one or not, there is associated an irreducible admissible representation π(θ) of G̃(F). Moreover, by Lemma 5.16 of [11], if ω lies in X(π̃) there is a character θ such that π̃ = π(θ). If G(ω) is the kernel of ω, regarded as a character of G̃(F), then π̃ restricted to G(ω) is the direct sum of two irreducible representations. Suppose T̃(F) is a Cartan subgroup corresponding to L, ψ a non-trivial character of F, and γ^{0} a regular element of T̃(F). According to Lemma 5.18 of [11] we may label these two components as π^{+}(θ) and π^{−}(θ) in such a way that

(2.4)  χ_{π^{+}(θ)}(γ) − χ_{π^{−}(θ)}(γ) = λ(L/F, ψ)ω (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) \frac{θ(γ) + θ(wγw^{−1})}{Δ(γ)}.

Here w lies in the normalizer of T̃(F) but not in T̃(F) and

Δ(γ) = |(γ_{1} − γ_{2})^{2}|^{1/2}/|γ_{1}γ_{2}|^{1/2}.

Let

θ̄(γ) = θ(wγw^{−1}) = θ(γ̄).

Observe in particular that π(θ_{1}) ≅ π̸(θ) unless θ_{1} = θ or θ_{1} = θ̄.

Suppose Π^{+}(θ) and Π^{−}(θ) are the components of the restrictions of π^{+}(θ) and π^{−}(θ) to G′; then a simple calculation shows that the map adjoint to

f → Φ^{T′}(f)

takes θ to

(2.5)  ∑_{π′∈Π^{+}(θ)} χ_{π′} − ∑_{π′∈Π^{−}(θ)} χ_{π′}.

If X(π̃) has r elements then there must be r − 1 non-isomorphic L and to each L a θ such that π̃ = π(θ). The span of the distributions (2.5) together with ∑ χ_{π′} where the sum is taken over all components of the restriction of π̃ to G′ is the same as the span of the χ_{π′}.

We say that two irreducible admissible representations π′_{1}, π′_{2} of G′ are L-indistinguishable if they both occur in the restriction of an irreducible, admissible representation of G̃(F) to G′ or, in other words, if π′_{2} is equivalent to h → π′_{1}(g^{−1}hg) for some g in G̃(F). This gives a partition of the equivalence classes of irreducible admissible representations of G′ into finite sets. If there is no L and θ such that π′ is a component of π(θ) or if the quadratic character corresponding to L is not trivial on G′, then the L-indistinguishable class of π′ consists of π′ alone. Otherwise it consists of two representations, or more precisely two equivalence classes, unless there are two different L_{1}, L_{2} as well as θ_{1}, θ_{2} such that G′ is contained in the kernel of ω_{1} and of ω_{2} and π′ is a component of both π(θ_{1}) and π(θ_{2}).

Indeed applying the Weyl integration to the formula (2.4) we see that

∑_{T′}([Ω(T′)] meas (Z′\T′))^{−1} ∫_{Z′\T′} |χ_{π^{+}(θ)}(γ) − χ_{π^{−}(θ)}|^{2}Δ^{2}(γ)dγ

is equal to 4 if

θ(γ) = θ(wγw^{−1})

for all γ in T′ and to 2 otherwise. Here Ω(T′) is the Weyl group of T′ in G′ and the sum is over the conjugacy classes of elliptic Cartan subgroups. Since wγw^{−1} corresponds to the conjugation γ → γ̄ we conclude from the orthogonality relations for square-integrable representations that if θ(γ) ≢ θ(γ̄) then the L-indistinguishable class of π′ consists of two elements if θ(γ) ≡ θ(γ̄) on T′ and of four if θ(γ) ≡ θ(γ̄) on T′.

In the latter case

γ → θ(γ/γ̄)

is not trivial but of order two. If B ⊆ F^{×} is

{Nm_{L/F} γ|θ(γ/γ̄) = 1}

then [F^{×} : B] = 4. If L_{1} = L, L_{2}, L_{3} are the three quadratic extensions with B ⊆ Nm L_{i} then it is easily seen that there are characters θ_{1} = θ, θ_{2}, θ_{3} of L_{i}^{×} such that the three representations

ρ_{i} = Ind(W_{L_{i}/F} , W_{L_{i}/L_{i}}, θ_{i})

of the Weil group are equivalent after inflation. Thus

π(θ_{1}) = π(θ_{2}) = π(θ_{3}).

Since [F^{×} : B] = 4 and there are only four classes L-indistinguishable from π(θ),

π(θ) ∼ ω ⊗ π(θ)

only if ω is trivial on B. Thus there cannot exist a further field L_{4} different from L_{1}, L_{2}, and L_{3} and a θ_{4} such that π(θ_{1}) ≃ π(θ_{4}).

If θ(γ) ≡ θ(γ̄) then π(θ) lies in the principal series and the L-indistinguishable class of π′ is easily seen to consist of two elements. In general, therefore, an L-indistinguishable class consists of 1, 2, or 4 elements.

As we have described them the sets Π^{+}(θ) and Π^{−}(θ) seem to depend on four choices, that of an additive character, a Cartan subgroup T′ corresponding to the field L, an isomorphism of T′ with

{x ∈ L^{×}|Nm x ∈ A}

and a regular element γ^{0}. However, the last three choices can be dispensed with, and it can be arranged that Π^{+}(θ) and Π^{−}(θ) depend only on ψ. This may not be significant. Given L we choose τ to lie in L but not in F and as above let

τ^{2} = uτ + v.

Take the imbedding of L^{×} in G̃(F) that assigns to a+bτ its matrix with respect to the basis {1, τ}, viz.

( \begin{matrix}a & bv \\ b & a + bu \end{matrix} ) .

This fixes T′ and the isomorphism. We take γ^{0} to be the image of the conjugate of τ. If we replace τ by x+yτ we replace T′ by g^{−1}T g where det g = y and (γ_{1}−γ_{2})/(γ_{1}^{0}−γ_{2}^{0}) by y^{−1}(γ_{1}−γ_{2})/(γ_{1}^{0}−γ_{2}^{0}). Consequently the sets Π^{+}(θ) and Π^{−}(θ) remain the same. According to the proof of Lemma 5.18 of [11], the elements of Π^{+}(θ) are the constituents of the representation π(θ, ψ) of Theorem 4.6 of [6].

It follows from the linear independence of characters that Π^{+}(θ) and Π^{−}(θ) are determined by the restriction θ′ of θ to T′; it is often convenient to write Π^{+}(θ′) and Π^{−}(θ′). We set

Π(θ′) = Π^{+}(θ′) ∪ Π^{−}(θ′).

If θ′_{1} and θ′_{2} are two characters of T′ and Π(θ′_{1}) and Π(θ′_{2}) are the same, then θ′_{1} and θ′_{2} extend to characters θ_{1} and θ_{2} of T̃(F) or L^{×} with π(θ_{1}) ≃ ω ⊗ π(θ_{2}). Here ω is a character of G′\G̃(F) or of A\F^{×}. If ω′(γ) = ω(Nm γ) then we may replace θ_{2} by ω′θ_{2} and suppose π(θ_{1}) ≃ π(θ_{2}). Then θ_{1} = θ_{2} or θ̄_{1} = θ_{2}. Consequently Π(θ′_{1}) = Π(θ′_{2}) if and only if θ′_{1} = θ′_{2} or θ̄′_{1} = θ′_{2}.

We have described the properties of the transform

f → Φ^{T′}(f)

for functions with compact support. If _{0}Z′ is a closed subgroup of the centre of G′ the transform may also be defined for functions which transform according to a character of _{0}Z′ and have support which is compact modulo _{0}Z′. It has similar properties, and they can be easily deduced from what we have already done (cf. [11]).

3. Stably invariant distributions. Two possible ways to define the notion of a stably invariant distribution on G′ present themselves. We could define a distribution to be stably invariant if it can be approximated by finite linear combinations of the distributions

f → ∑_{δ∈𝔇(T′)} Φ^{δ}(γ, f)

with γ regular in T′ or, more naively, if it is invariant under conjugation by elements of G̃(F). The two possibilities are likely to be equivalent, but we do not trouble ourselves about this, and simply choose the second, because it is easier to work with. In this paragraph we shall examine a number of distributions that arise in the trace formula to see how far they depart from stable invariance.

Let A′ be the group of diagonal matrices in G′, let

N(F) = \{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) |x ∈ F \} ,

and let η be a character of A′. We do not assume that η has absolute value 1. We introduce the representation

ρ(η) : g → ρ(g, η)

of G′ on the space of smooth functions φ on N(F)\G′ satisfying

φ(ag) = η(a) |\frac{α}{β}|^{1/2} φ(g), a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ∈ A′.

If f is a smooth function on G′ with compact support set

ρ(f, η) = ∫_{G′} f(g)ρ(g, η)dg.

Lemma 3.1. The distribution

f → trace ρ(f, η)

is stably invariant.

If g ∈ G̃(F) we set

gf(h) = f(g^{−1}hg).

We have to show that

trace ρ(^{g}f, η) = trace ρ(f, η),

but it is enough to do this when g is diagonal because G̃(F) = A(F)G′. Then

R_{a} : φ → ^{a}φ

maps the space on which ρ(η) acts to itself. Since

ρ(^{a}f, η)R_{a} = R_{a}ρ(f, η)

and

R_{a}^{−1} = R_{a^{−1}}

the lemma is clear.

If F is non-archimedean

K′ = G′ ∩ G̃(O_{F})

where O_{F} is the ring of integers in F. Otherwise let K′ be the group of orthogonal or unitary matrices in G′. Let A′ be the group of diagonal matrices in G′. Every element of G′ may be written as a product g = nak with k ∈ K′,

a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) ∈ A′,

and

n = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) .

Set β(g) = |α/β| and λ(g) = β(g) + β(wg) with

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

Then λ(g) = β(wn). If γ lies in A′ and has distinct eigenvalues α, β set

Δ(γ) = |α − β|/|αβ|^{1/2}

and introduce two distributions

\begin{aligned}F (γ, f) & = Δ(γ) ∫_{A′\G′} f(g^{−1}γg)dg, \\ A_{1}(γ, f) & = Δ(γ) ∫_{A′\G′} f(g^{−1}γg) ln λ(g)dg. \end{aligned}

In both cases the integral may be replaced by one over Ã(F)\G̃(F) because A′\G′ = Ã(F)\G̃(F). The first distribution is clearly stably invariant.

The second is not even invariant. However, it is invariant under G̃(O_{F}). Since A′\G′ is equal to Ã(F)\G̃(F) we may regard the space of functions on which ρ(η) acts as a space of functions on G̃(F). We extend η to a character η̃ of Ã(F) and treat ρ(η) as the restriction of ρ(η̃). We may identify the space of ρ(η̃) with a space of functions on G̃(O_{F}), for the functions on which ρ(η̃) acts are determined by their values on G̃(O_{F}). The space of functions is the same for η̃ and for

η̃_{s} : a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) → η̃(a) |\frac{α}{β}|^{s};

and this will allow us to introduce derivatives with respect to s. If g ∈ G̃(F) let kg = n_{1}a_{1}k_{1} and let N(g) be the operator on the space of ρ(η̃) given by multiplication by ln β(a_{1}). We introduce dual measures on A′ and on D^{0}, its Pontrjagin dual. The kernel of ρ(f, η) is

K_{η}(k_{1}, k_{2}) = ∫_{A′} ∫_{N(F)} f(k_{1}^{−1}ank_{2})λ(a)^{1/2}dadn

provided the measure on K′ is so chosen that

∫_{G′} f(g)dg = ∫_{A′} ∫_{N(F)} ∫_{K′} f(ank)dadndk.

The Fourier transform of

∫_{K′} K_{η}(k, k)dk

is easily seen to be F (γ, f). The trace of ρ(f, η)N(g) is

∫_{K′} K_{η}(k, k) ln β(a_{1})dk.

Taking the Fourier transform with respect to η we obtain

∫_{N(F)} ∫_{K′} f(k^{−1}ank)β(a)^{1/2} ln β(a_{1})da′dndk′.

Lemma 3.2. The difference

A_{1}(γ,^{g^{−1}}f) − A_{1}(γ, f)

is the sum of the Fourier transform of

trace ρ(f, η)N(g)

at γ and γ̃ where

γ̃ = ( \begin{matrix}β & 0 \\ 0 & α \end{matrix} )

if

γ = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) .

The difference is equal to the sum of

Δ(γ) ∫_{_{A}(F)\_{G}(F)} f(h^{−1}γh){ln β(hg) − ln β(h)}dh

and

Δ(γ) ∫_{_{A}(F)\_{G}(F)} f(h^{−1}γh){ln β(whg) − ln β(wh)}dh.

Replacing h by w^{−1}h in the second integral, we see that it is enough to show that the first integral is the value of the Fourier transform of trace ρ(f, η)N(g) at γ. However, if h = nak then hg = nan_{1}a_{1}k_{1} and ln β(hg) − ln β(n) = ln β(a_{1}). Standard manipulations complete the proof.

There are still more distributions to be investigated. Let L(1+s, 1_{F}) be the local zeta-function of F at 1 + s. If a lies in Z′ the centre of G′ and

n_{a} = a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} )

set

θ(a, s, f) = \frac{1}{L(1 + s, 1_{F})} ∫_{_{G_{n}}(F)\_{G}(F)} f(g^{−1}n_{a}g)β(g)^{−s}dg.

We suppose that the measure on G̃_{n}(F), the centralizer of n, is that associated to the form

\frac{dz}{z} dx

if a typical element of G̃_{n} is

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

and that the measure on Ã(F) is that associated to

\frac{dα}{α} \frac{dβ}{β}

when the typical element is

( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) .

Lemma 3.3. The distribution

\frac{d}{ds}θ(a, 0,^{g^{−1}} f) − \frac{d}{ds}θ(a, 0, f)

is −1 times the Fourier transform of

trace ρ(f, η)N(g)

at a divided by L(1, 1_{F}).

When we take the derivative and then the difference, the term involving the derivative of the L-function drops out and we are left with

\frac{−1}{L(1, 1_{F})} ∫_{_{G_{n}}(F)\_{G}(F)} f(h^{−1}n_{a}h)\{ln β(hg) − ln β(h)\}dh

or

\frac{−1}{L(1, 1_{F})} ∫_{_{G_{n}}(F)\_{G}(F)} f(h^{−1}an_{a}h) ln β(a_{1})dh

if a_{1} is defined as above. However, the integral may be written

∫_{_{Z}(F)\_{A}(F)} ∫_{_{G}(O_{F})} f(k^{−1}at^{−1}n_{a}tk) ln β(a_{1})λ(t)dtdk.

Changing variables we see that this equals

∫_{N(F)} ∫_{G(O_{F})} f(k^{−1}ank) ln β(a_{1})dndk

which is, because of the normalization of measures, also

∫_{N(F)} ∫_{K′} f(k^{−1}ank) ln β(a_{1})dndk.

The next distributions to be considered are defined by intertwining operators. We associate to η the character

μ : α → η (( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ))

of F^{×} and consider the normalized intertwining operators R(η) on the space of ρ(η) or ρ(η̃)

R(η)φ(g) = ϵ(0, μ, ψ)\frac{L(1, μ)}{L(0, μ)} ∫_{N(F)} φ(wng)dn.

Here ψ is some non-trivial additive character of F and ϵ(0, μ, ψ) is the usual factor. The measure on N(F) is that associated to the form dx and the character ψ. The integrals converge when |μ(ω̃)| < 1 and the intertwining operator can be defined by analytic continuation for |μ(ω̃)| = 1. Here ω̃ is a uniformizing parameter for F. When |μ(ω̃)| = 1 the operator R(η) is invertible. Lemma 3.4. The difference

trace R^{−1}(η)R′(η)ρ(^{g^{−1}}f, η) − trace R^{−1}(η)R′(η)ρ(f, η)

is equal to

trace ρ(f, η)N(g) + trace ρ(f, η_{1})N(g).

The prime denotes differentiation with respect to the parameter s. It will be enough to show that

ρ(g, η̃_{1})R′(η)ρ(g, η̃)^{−1} − R′(η) = R(η)N(g) + N(g)R(η).

Here η̃_{1} is defined by

η̃_{1} (( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )) = η̃(( \begin{matrix}β & 0 \\ 0 & α \end{matrix} ))

and η_{1} has been defined in a similar fashion. Because we can invoke the principle of permanence of functional relations, it suffices to verify the desired relation when |μ(ω̃)| < 1 and the intertwining operator is defined by the integral. In the difference the derivative of the L-functions disappear and it is therefore adequate to verify the analogous relation for the unnormalized intertwining operators

M(η)φ(g) = ∫_{N(F)} φ(wng)dn.

Then

M′(η)φ(k) = ∫_{N(F)} φ′(wnk)dn

where in the integrand it is understood that we start from a fixed function on G̃(O_{F}) and then extend it to a function in the space of ρ(η) or ρ(η̃). The extended function will then depend on the parameter s locally, and the prime indicates that the derivative with respect to s has been taken.

If wnk = η_{1}a_{1}k_{1} the integrand is equal to ln β(a_{1})φ(wnk). If kg = n_{2}a_{2}k_{2} and wnk_{2} = n′_{1}a′_{1}k′_{1} then

ρ(g, η̃_{1})M′(η)ρ(g^{−1}, η̃)φ(k) = ∫ η̃_{1}(a_{2})β^{1/2}(a_{2}) ln β(a′_{1})φ(wnk_{2}g^{−1})dn.

Since

k_{2}g^{−1} = a_{2}^{−1}n_{2}^{−1}k,

we may change variables in the integrand, replacing n by a_{2}^{−1}nn_{2}a_{2} to obtain

∫ ln β(a′_{1})φ(wnk)dn

where a′_{1} is now defined by

wa_{2}^{−1}nn_{2}a_{2}k_{2} = n′_{1}a′_{1}k′_{1}.

But the left side is wa_{2}^{−1}w^{−1}wnkg. If k_{1}g = n_{3}a_{3}k_{3} then

a′_{1} = wa_{2}^{−1}w^{−1}a_{1}a_{3}

and

ln β(a′_{1}) = ln β(a_{2}) + ln β(a_{1}) + ln β(a_{3}).

Since

M(η)N(g)φ(k) = ∫_{N(F)} ln β(a_{3})φ(wnk)dn

and

N(g)M(η)φ(k) = ∫_{N(F)} ln β(a_{2})φ(wnk)dn

the lemma follows.

Suppose χ is a given character of _{0}Z′, a closed subgroup of the centre of G′. If f is a function which is smooth and compactly supported modulo _{0}Z′ and satisfies

f(zg) = χ(z)f(g), z ∈ _{0}Z′,

and if η is a character of A′ whose restriction to _{0}Z is χ^{−1} then we may define ρ(f, η) to be

∫_{_{0}Z′\G′} f(g)ρ(g, η)dg.

We can also define distributions for the class of these functions and carry out a completely analogous discussion.

There are two more lemmas about intertwining operators that we will need for our treatment of the trace formula. If η = η_{1} then R(η) intertwines the representation ρ(η) with itself.

Lemma 3.5. If η can be extended to a character η̃ of Ã(F) which satisfies η̃_{1} = η̃, then R(η) is the identity.

This is basically the first part* of Lemma 5.7 of [11]. If η = η_{1} but η does not extend to an η̃ with η̃ = η̃_{1} then the character μ of F^{×} associated to η by

μ(α) = η (( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ))

is quadratic but not trivial and hence defines a quadratic extension L. We associate to L a Cartan subgroup T̃ of G̃ with T̃(F) ≃ L^{×}. Its intersection with G′ is then a Cartan subgroup of G′ and

T′ ≃ {x ∈ L^{×} | Nmx ∈ A}.

Define a character θ of T′ by

θ(x) = η (( \begin{matrix}Nmx & 0 \\ 0 & 1 \end{matrix} )) .

The elements of Π(θ) are the irreducible constituents of ρ(η). Since we have chosen the additive character ψ we can speak of Π^{+}(θ) and Π^{−}(θ).

* Observe that Section 5 of the notes [11] became Section 7 when they were published.

Lemma 3.6. The trace

trace R(η)ρ(f, η)

is equal to

∑_{π∈Π^{+}(θ)} χ_{π}(f) − ∑_{π∈Π^{−}(θ)} χ_{π}(f).

Observe that the previous lemma could be formulated in the same way if we took T′ to be the split Cartan subgroup, Π^{−}(θ) to be empty, and Π^{+}(θ) to be the constituents of ρ(η). Lemma 3.6 will be proved in much the same way as Lemma 5.8 of [11]. Notice how important the choice of normalization is! According to the discussion of [11], we replace R(η) by μ(a)R(η) if we replace ψ(x) by ψ(ax). If μ(a) = −1 this is compensated by the interchange of Π^{+}(θ) and Π^{−}(θ).

It is certainly enough to prove the lemma when

G′ = G(μ) = {g | μ(det g) = 1}.

Then Π^{+}(θ) consists of a single element π^{+} and Π^{−}(θ) of π^{−} and

ρ(η̃) = π^{+} ⊕ π^{−}.

Suppose first that F is non-archimedean. If we take the Kirillov model of ρ(η) then its space V is the sum of the space V^{+} of functions in it with support on Nm L^{×} and the space V^{−} of functions in it with support in F^{×} − Nm L^{×}. π^{+} acts on V^{+} and π^{−} on V^{−}. Since R(η) commutes with G′ it is multiplication by a function a + bμ(α). Here a and b are constants and α is a variable in F^{×}.

If η̃ is an extension of η to A(F̃) then η̃ has the form

η̃(( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )) = ν(αβ)μ(β)

and R(η̃) is equal to R(η). We are going to apply formula (1) of §5 of [11], taking account of the differences in notation. If

ψ(α) = W (( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} ))

is a function in V then

ψ(α) ∼ cν(α)|α|^{1/2}{φ(1) + μ(α)R(η)φ(1)}.

Here c is a constant, namely

μ(−1)ϵ(0, μ, ψ).

As we observed the effect of applying R(η) is to multiply by a + bμ(α). On the other hand the effect on the asymptotic expression above is to change it to

cν(α)|α|^{1/2}{R(η)φ(1) + μ(α)R^{2}(η)φ(1)}.

Lemma 5.8 of [11] implies that R^{2}(η) is the identity. We conclude that

\begin{aligned}aφ(1) + bR(η)φ(1) & = R(η)φ(1) \\ bφ(1) + aR(η)φ(1) & = φ(1) \end{aligned}

and hence that

aI + (b − 1)R(η) = 0.

The operator R(η) cannot be a scalar because η̃_{1} ≠ η̃. We conclude that a = b−1 = 0. The lemma follows.

If F is archimedean then F is R; and the lemma is proved by evaluating the appropriate definite integrals. This is straightforward. Since it is possible, at the cost of a little additional effort, to manage without the lemma for archimedean fields, and indeed to deduce it from the global considerations, we omit the calculations.

We should remark at some point and we do it now that if θ is unramified and O_{F} is the largest ideal on which ψ is trivial then there is exactly one element of Π(θ) that contains the trivial representation of K′ and it lies in Π^{+}(θ). This can be seen by examining the constructions of [6].

4. Quaternion algebras. We now let G̃(F) be the multiplicative group D^{×} of a quaternion algebra D over F and, choosing A as before with A ⊆ Nm D^{×}, define G′ to be

{g ∈ G̃(F) | Nmg ∈ A}.

A Cartan subgroup of T′ of G′ is again of the form G′ ∩ T̃(F) where T̃ is a Cartan subgroup of G̃ over F. T′ or T̃ correspond to a quadratic extension L of F. Set

\begin{aligned}𝔈(T′) & = F^{×}/ANmL^{×} \\ 𝔇(T′) & = Nm D^{×}/ANmL^{×}. \end{aligned}

If F is real 𝔇(T′) is a proper subset of 𝔈(T′).

If γ in T′ is regular and δ in 𝔇(T′) is represented by h in G̃(F) we set

Φ^{δ}(γ, ξ) = ∫_{h^{−1}T′h\G′} f(g^{−1}h^{−1}γhg)dg

as before and define

Φ^{T′/κ′}(γ, f) = d(γ, κ′) ∑_{𝔇(T′)} κ′(δ)Φ^{δ}(γ, f).

κ′ is a character of 𝔈(T′). If κ′ is not trivial we define

d(γ) = d(γ, κ′)

as in §2 and set

Φ^{T′}(γ, f) = Φ^{T′/κ′}(γ, f).

Let Z′ be the centre of G′ and extend Φ^{T′}(f) to all of T′ by the equation

Φ^{T′}(a, f) = 0 a ∈ Z′.

Lemma 4.1. The function Φ^{T′}(f) is smooth.

Since Z′\G′ is compact the function γ → Φ^{δ}(γ, f) is given by

\frac{1}{meas (Z′\T′)} ∫_{Z′\G′} f(g^{−1}h^{−1}δhg)dg

and is defined and smooth on all of T′. If F is R then d(γ) is also smooth on all of T′ and 0 on Z′. If F is non-archimedean then 𝔇(T′) = 𝔈(T′) and

∑_{𝔇(T′)} κ′(δ)Φ^{δ}(γ, f)

is 0 near Z′.

The dual map on distributions takes the character θ′ of T′ to a function Θ which satisfies

Θ(h^{−1}gh) = κ′(δ)Θ(g),

if h ∈ G̃(F) represents δ, and which is supported on the union of the Cartan subgroups stably conjugate to T′. Let ω be the quadratic character of F^{×} associated to L, so that ω(Nmh) = κ′(δ) and let the standard involution in D be denoted by a bar. If w in G̃(F) satisfies w^{−1}γw = γ̃ for γ in T′ then ω(Nmw) = −ω(−1). Formal manipulations then establish that on T′ the function Θ is given by

Θ(γ) = λ(L/F, ψ)ω (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) \frac{θ′(γ) − θ′(w^{−1}γw)}{Δ(γ)}.

In particular Θ is identically 0 if θ′(γ) ≡ θ′(γ̃), and when F is non-archimedean is 0 near Z′.

If θ̃ is an extension of θ′ to T̃(F) and π(θ̃) the associated irreducible admissible representation of G̃(F), let Π(θ′) be components of the restriction of π(θ̃) to G′. If θ̃(γ) ≡ θ̃(γ̃) then π(θ̃) does not exist, but it does otherwise, and if F = R the set Π(θ′) consists of a single element π. It is easily seen that

χ_{π} = ±Θ.

As in §2 we say that two irreducible admissible representations π_{1}, π_{2} of G′ are L-indistinguishable if π_{2} = π_{1}^{g} with g ∈ G̃(F). It is clear that π_{1} and π_{2} are L-indistinguishable if and only if there is a representation π̃ of G̃(F) whose restriction to G′ contains both π_{1} and π_{2}. In general

π̃ | G′ ≃ ⊕_{i=1}^{τ}cπ_{i}

where c = c(π̃) = c(π_{i}) is a positive integer and the sum is over an L-indistinguishable class. The number c may not be 1. In order to discover something about it we compare representations of G̃(F) with representations of GL(2, F). We now denote GL(2) by H and change the notation of §2 accordingly. Thus

H′ = {h ∈ GL(2, F) | det h ∈ A}.

Let τ̃ be the representation of H̃(F) associated to π̃ and let

τ̃ | H′ = ⊕_{i=1}^{s}τ_{i}.

With no loss of generality we may assume that the central character of π̃ has absolute value 1. The character χτ̃ of τ̃ on the elliptic elements is the negative of χπ̃, and χπ̃ restricted to G′ and χτ̃ restricted to H′ are stable. The orthogonality relations on H′ state that

\frac{1}{2} ∑_{U′} \frac{[𝔇(u′)]}{meas Z′\U′} ∫_{Z′\U′} |χτ̃(γ) |^{2} Δ^{2}(γ)dγ = s.

The sum is over a set of representations for the stable conjugacy classes of elliptic Cartan subgroups. If F is non-archimedean the orthogonality relations on G′ state that

\frac{1}{2} ∑_{T′} \frac{[𝔇(T′)]}{meas Z′\T′} ∫_{Z′\T′} |χπ̃(γ)|^{2}Δ^{2}(γ)dγ = rc^{2}.

The sum is over a set of representatives for the stable conjugacy classes of Cartan subgroups. If F is non-archimedean and T′ and U′ are Cartan subgroups corresponding to the same quadratic extension then

[𝔇(T′)] = [𝔇(U′)].

If F is R then [𝔇(T)] is always 1 while [𝔇(U′)] is 1 or 2, but then s = [𝔇(U′)]. The conclusion is that rc^{2} = s when F is non-archimedean and that rc^{2} = 1 when F is R. Thus when F is R the integers r and c are both 1. When F is non-archimedean s is 1, 2, or 4. If s is 1 or 2 then r = s and c = 1. If s is 4 either r = 4 and c = 1 or r = 1 and c = 2. We shall see eventually that only the latter possibility occurs, but we will need the help of the trace formula.

In order to apply the trace formula we shall need to compare distributions on G′ and on H′. We set

Φ^{T′/1}(γ, f) = ∑_{𝔇(T′)} Φ^{δ}(γ, f).

If we agreed that Φ^{δ}(γ, f) = 0 for δ in 𝔈(T′) but not in 𝔇(T′) we could also write the sum on the right as

∑_{𝔈(T′)} Φ^{δ}(γ, f).

It is a stably invariant distribution and does not depend on the choice of T′ within a stable conjugacy class. If we replace T′ by h^{−1}T′h and γ by h^{−1}γh with h in G̃(F) we obtain the same distribution. T′ determines a stable conjugacy class {U′} of H′ and an isomorphism ψ : T′ →^{∼} U′, determined up to stable conjugation. If ϕ is a smooth function on H′ with compact support we may also introduce

Φ^{U′/1}(ζ, ϕ) = ∑_{D(U′)} Φ^{δ}(ζ, ϕ).

It follows easily from Lemma 4.1 of [11] that if f is given there is a ϕ satisfying

Φ^{U′/1}(ζ, ϕ) ≡ 0

if U′ is split and

Φ^{U′/1}(ζ, ϕ) = Φ^{T′/1}(γ, f)

if T′ and U′ are corresponding tori and ζ = ψ(γ). The adjoint map is only defined on stably invariant distributions.

Suppose T is an L-indistinguishable class for H′. If the elements of T belong to the principal series then

∑_{τ∈T} χ_{τ} → 0.

However, if the elements of T belong to the discrete series and Π is the corresponding class for G′ then

∑_{τ∈T} χ_{τ} → − ∑_{π∈Π} c(π)χ_{π}.

If τ is one-dimensional and π is the corresponding one-dimensional representation of G′ then χ_{τ} → χ_{π}.

5. The trace formula. F will now be a global field. Let A be a closed subgroup of I_{F} = GL(1, A_{F}) of the form

A = ∏_{v} A_{v}.

The product is over all places of F and A_{v} is a closed subgroup of GL(1, F_{v}). We suppose in addition that F^{×}A is closed. If B is an open subgroup of A and A_{F} = A ∩ F^{×} we also demand that [A : A^{2}A_{F} B] < ∞. Set

G′ = {g ∈ G̃(A_{F}) | det g ∈ A}.

Let _{0}Z′ be a closed subgroup of the centre Z′ of G′, with _{0}Z′F^{×} closed,

0Z′ = ∏_{ν} _{0}Z′_{ν},

and

0Z′Z′_{F} \Z′, Z′_{F} = Z′ ∩ F^{×}

compact. Let χ be a character of _{0}Z′ trivial on _{0}Z′_{F} = _{0}Z′ ∩ F^{×} and of absolute value one.

We want to apply the trace formula to the space of measurable functions φ on G′_{F} \G′, with G′_{F} = G̃(F) ∩ G′, which satisfy

\begin{matrix}(i) φ(zg) = χ^{−1}(z)φ(g), z ∈ _{0}Z′, \\ (ii) ∫_{_{0}Z′G′_{F} \G′} |φ(g)|^{2}dg < ∞. \end{matrix}

This is not the exact context in which a detailed proof has been published, but that may be in the nature of things, for it is a principle that Selberg discovered, more than a formula, and principles when they are effective are also plastic, and do not admit a definitive form. To carry out the verification of the formula with the minor modifications now required would not however be very profitable, for it would amount to little more than a transcription of [6]. We content ourselves with stating the result.

The space of functions φ is a direct sum of two subspaces

L′(G′_{F} \G′, χ) ⊕ L′′(G′_{F} \G′, χ)

both invariant under the action of G′. The representation of G′ on the first subspace is a continuous direct integral of irreducible representations and is constructed by the Eisenstein series, while L′′(G′_{F} \G′, χ) is a direct sum of irreducible representations. Let r be the representation of G′ on it. Suppose

f : g → Πf_{v}(g_{v})

is a function on G′. We suppose that

f(zg) = χ(z)f(g), z ∈ _{0}Z′,

and that its support is compact modulo _{0}Z′. Moreover, each f_{v} is to be smooth and for almost all v, f_{v} is to be supported on _{0}Z′_{v}(G′ ∩ G̃(O_{F_{v}})) and the relation

meas (_{0}Z′_{v}\_{0}Z′_{v}(G′ ∩ G̃(O_{F_{v}}))f(zk) = χ(z), z ∈ _{0}Z′_{v}, k ∈ G′ ∩ G̃(O_{F_{v}}),

is to be satisfied. We define

r(f) = ∫_{_{0}Z′\G′} f(g)r(g)dg.

The operator r(f) is of trace class and the trace formula provides a complicated but useful expression for it.

This expression is usually presented as the sum of several parts ([1], [3], [6], [11]). The authors of [6] do not seem to have been able to keep a firm grip on the constants that arose in their discussion of the trace formula. We timorously suggest the following corrections: p. 516, line 2*, replace c by c/4; p. 531, divide the second, third, and fourth displayed expressions by 2; p. 540, lines 7, 9, and 11, replace c_{1}^{2s} and c_{1}^{2t} by c_{1}^{s} and c_{1}^{t}. The last change has then to be made in the ensuing calculations as well. However, we are going to state the formula in a slightly different situation, and will have the opportunity to make new errors all of our own. The first part is

(5.1)  ∑_{γ∈_{0}Z′_{F} \Z_{F′}} f(γ).

The second is a sum over the elliptic conjugacy classes {γ} in (_{0}Z′ ∩ F^{×})\G′_{F} of

(5.2)  δ(γ)^{−1} meas (_{0}Z′G′_{F} (γ)\G′(γ)) ∫_{G′(γ)\G′} f(g^{−1}γg)dg.

Here G′(γ) and G′_{F} (γ) are the centralizers of γ in G′ and G′_{F} and δ(γ) is the index of_{0}Z′\_{0}Z′G′_{F} (γ) in the centralizer of γ in _{0}Z′\_{0}Z′G′_{F}. It is 1 or 2.

As before, let A′ be the group of diagonal matrices in G′ and let

K′_{v} = G′ ∩ G̃(O_{F_{v}}).

It is understood that G̃(O_{F_{v}}) is to be the group of orthogonal or unitary matrices when v is real or complex. Let

A^{0} = \{( \begin{matrix}a & 0 \\ 0 & β \end{matrix} ) ∈ A′ | |α| = |β|\} .

The group A^{0}\A′ is isomorphic to R by means of

( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) → x = ln |\frac{α}{β}| .

We choose a measure on _{0}Z′\A′ composed of the measure dx on A^{0}\A′ and a measure on _{0}Z′\A^{0} for which

meas _{0}Z′A′_{F} \A^{0} = 1.

Here A′_{F} is A′ ∩ A(F). We may also suppose that this measure is given as a product measure. The next term of the trace formula is then

(5.3)  −\frac{λ}{2} ∑_{v} \begin{matrix}∑ & ′ \\ γ∈_{0}Z′_{F} \A′_{F} \end{matrix} \frac{A_{1}(γ, f_{v})}{L(1, F_{v})} ∏_{w≠v} \frac{F (γ, f_{w})}{L(1, f_{w})}

with _{0}Z′_{F} = _{0}Z′ ∩ F^{×}. Here λ is the residue of the global L-function L(s, 1_{F}) at s = 1. The factor λ and the denominators appear because of the relation between local and global Tamagawa measures.

On

N(A) = \{( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} ) | x ∈ A\}

we choose a product measure for which

meas (N(F)\N(A)) = 1.

The next part of the trace formula appears at first in a garb that conceals the features of concern to us. It is the constant term of the Laurent expansion of

(5.4)  ∑_{a∈_{0}Z′_{F} \Z′_{F}} ∑_{x∈F^{×^{2}} A_{F} \F^{×}} meas (_{0}Z′Z′_{F} \Z′) ∫_{Z′N(A)\G′} f (g^{−1}a ( \begin{matrix}a & x \\ 0 & 1 \end{matrix} ) g) λ(g)^{−s}dg

at s = 0. Here A_{F} = A ∩ F^{X} and if

g = ( \begin{matrix}1 & y \\ 0 & 1 \end{matrix} ) ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) k, k ∈ K′ = ∏ K′_{v},

then

β(g) = |α/β|.

Suppose D^{0} is the set of all characters of A′_{F} \A′ which equal χ^{−1} on_{0}Z′. Another contribution to the trace formula is the sum over all η in D^{0} for which η = η_{1} of

(5.5)  −\frac{1}{4} trace (M(η)ρ(f, η)).

Recall that

η_{1} (( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )) = η (( \begin{matrix}β & 0 \\ 0 & α \end{matrix} )) .

There are two more contributions. D^{0} is the union of connected components, the component containing η^{0} consisting of those η of the form

η : a = ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) → η^{0}(a) |\frac{α}{β}|^{s}

with s purely imaginary. The dual measure on D^{0} is 1/2π|ds|. If

μ = μ_{η} : α → η (( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} ))

then μ is a character of the idèle class group. We set

m(η) = L(1, μ^{−1})/L(1, μ).

One contribution is

(5.6)  \frac{1}{4π} ∫_{D^{0}} m(η) trace ρ(f, η)|ds|.

Another is

∑_{v} \frac{1}{4π} ∫_{D^{0}}\{trace R^{−1}(η_{v})R′(η_{v})ρ(f_{v}, η_{v})\} \{ ∏_{w≠v} trace ρ(f_{w}, η_{w})\} |ds|.

The representations ρ(·, η_{v}) have been defined in the third paragraph.

The distribution

f → trace r(f)

will not be stable. Our purpose is to write it as the sum of a stable term and a term that can be analyzed by means of the stabilized trace formula for groups of lower dimension and the principle of functoriality. The group G = SL(2) is very special, and for it the lower dimensional groups are just Cartan subgroups, but one of our purposes is to illuminate the definitions of [12], and so we begin with them in mind, with G′ = G(A), and with _{0}Z′ = 1.

Two elements of G(F) will be called stably conjugate* if they are conjugate in G(F̄) or, what is the same, in G̃(F). We take the expression (5.2) and sum over the conjugacy classes within the stable conjugacy class of γ. If we introduce the global form of the notation of the second paragraph, this sum may be written

(5.7)  meas (T (F)\T (A)) ∑_{𝔇(T/F)} Φ^{δ}(γ, f)

with T = G(γ). With the present assumptions δ(γ) is 1.

If 𝔇(T/A) is the direct sum of the 𝔇(T/F_{v}) we have a map

𝔇(T/F) → 𝔇(T/A).

There are a number of simplifying factors in the present circumstances. First of all

𝔇(T/F) = 𝔈(T/F) = H^{1}(F, T)

and

D(T/A) = 𝔊(T/A) = ⊕H^{1}(F_{v}, T).

Moreover

H^{1}(F, T) → ⊕H^{1}(F_{v}, T)

is injective. If L is the quadratic extension defined by T and C_{L} the idèle class group of L then

(5.8)  H^{1}(F, T) → ⊕H^{1}(F_{v}, T) → H^{1}(𝔈(L/F), X_{∗}(T) ⊗ C_{L})

is exact. Since T is not split the Tate-Nakayama theory shows that the last group is

H^{−1}(𝔊(L/F), X_{∗}(T)) = X_{∗}(T)/{Σσλ − λ | λ ∈ X_{∗}(T), σ ∈ 𝔊(L/F)}.

There are analogues of these statements for general groups. A special feature of G = SL(2) is that the second arrow of (5.8) is surjective. Thus the dual of 𝔈(T/F)\𝔈(T/A) is isomorphic to the set of 𝔊(L/F)-invariant homomorphisms κ of X_{∗}(T) into C^{×}.

The sum (5.7) is equal to

meas (T (F)\T (A))[𝔈(T/A) : 𝔈(T/F)]^{−1} ∑_{κ} ∑_{𝔈(T/A)} κ(δ)Φ^{δ}(γ, f).

* Prudence must be exercised when transferring this notion to other groups.

According to the definitions of [12], a group H is associated to the pair T, κ. If κ is trivial this group is just the quasi-split form of G, namely G itself, and

f → ∑_{𝔈(T/A)} Φ^{δ}(γ, f)

is a stable distribution.

If κ is the non-trivial homomorphism, and there is only one of them, the group H is T. If w is a divisor of v in L then by the Tate-Nakayama theory

H^{1}(F_{v}, T) = {λ ∈ X_{∗}(T)| ∑_{𝔊(L_{w}/F_{v})} σλ = 0}/{Σσλ − λ|λ ∈ X_{∗}(T), σ ∈ 𝔊(L_{w}/F_{v})}

and the map

H^{1}(F_{v}, T) → H^{1}(𝔊(L/F), X_{∗}(T) ⊗ C_{E})

is simply λ → λ. In particular κ determines a homomorphism κ_{v} of 𝔇(T/F_{v}) into C^{×}. It is trivial if v splits and non-trivial otherwise. Choose γ^{0} in T̃(F) and a non-trivial character of F \A. Then

∏_{v} λ(L_{v}/F_{v}, ψ_{v})κ_{v} (\frac{γ_{1} − γ_{2}}{γ_{1}^{0} − γ_{2}^{0}} ) \frac{|(γ_{1} − γ_{2})^{2}|_{v}^{1/2}}{|γ_{1}γ_{2}|_{v}^{1/2}} = 1

and

∑_{𝔊(T/A)} κ(δ)Φ^{δ}(γ, f) = Π_{v}Φ^{T/κ_{v}}(γ, f_{v})

and

meas (Z′T (F)\T (A))Π_{v}Φ^{T/κ_{v}}(γ, f_{v})

is one term of the trace formula for the abelian pair Z′T (F)\T (A) and the function

γ → Π_{v}Φ^{T/κ_{v}}(γ, f_{v}) = Φ^{T/κ}(γ, f)

on T (A).

For each isomorphism class of quadratic extension we choose a torus T, in other words we choose a set of representatives for the stable conjugacy classes of non-split tori. We may choose the representatives γ of the stable conjugacy classes arising from (5.2) to lie in one of these T. The stable conjugacy class determines T, but within T there are two possible representatives for the stable conjugacy classes, and so we must divide by 2 if we want to sum over all regular elements in T (F). Since the index [𝔈(T/A) : 𝔈(T/F)] is also 2, the total unstable contribution from (5.2) is \frac{1}{4} times the sum over T of

(5.9)  meas (T (F)\T (A)) ∑′ Φ^{T/κ}(γ, f).

The prime indicates that we only sum over the regular elements γ of T (F). We almost have what we were seeking. We must now hope that when we remove the stable part from the remaining terms of the trace formula we will be left with the summands missing from (5.9), namely

\frac{1}{4} ∑_{T} meas (T (F)\T (A)) ∑_{γ∈Z(F)} Φ^{T/κ}(γ, f).

Here Z is the centre of G and Z(F) consists of two elements.

We have preferred to treat not merely the group G(A) but the groups G′ as well, and we carry out the complete discussion for them. We examine the terms (5.2) once again. The group G′_{F} (γ) is the intersection T′_{F} of G′_{F} with a torus T̃ in G̃. If T̃ is associated to the quadratic extension L we define

𝔈(T′/F) = F^{×}/A_{F} Nm_{L/F} L^{×}, A_{F} = A ∩ F^{×},

and

𝔈(T′/A) = I_{F} /ANm_{L/F} I_{L},

where I_{F} and I_{L} are the idèles of F and L. It is also useful to introduce the subsets 𝔇(T/F) and 𝔇(T/A) whose elements can be realized as norms from G̃, but only for groups defined by quaternion algebras is this necessary. The natural map from 𝔈(T′/F) to 𝔈(T′/A) has a cokernel of order 1 or 2. Let μ be the order of its kernel.

If the cokernel is trivial and we sum over the stable conjugacy classes within the conjugacy class of γ we obtain

μδ(γ)^{−1} meas (_{0}Z′G′_{F} (γ)\G′(γ)) ∑_{δ∈𝔇(T′/A)} Φ^{δ}(γ, f),

which is a stable distribution. If it is not trivial the contribution of the stable conjugacy class of γ to the trace formula is \frac{1}{2} the sum of this expression and

μδ(γ)^{−1} meas (_{0}Z′G′_{F} (γ)\G′(γ)) ∑_{𝔇(T′/A)} κ′(δ)Φ^{δ}(γ, f).

Here κ′ is the non-trivial character of Im(T′/A)/𝔈(T′/F). The stable conjugacy class of γ has two representatives within a given T′ modulo _{0}Z′ if and only if δ(γ) = 1. Thus if we choose a set of representatives T′ for the stable conjugacy classes of Cartan subgroups with

[𝔈(T′/A) : Im 𝔈(T′/F)] = 2

the total labile contribution to the trace formula of the sum over the terms (5.2) is

(5.10)  \frac{1}{4} ∑_{T′} μ meas (_{0}Z′T′_{F} \T′) ∑′ Φ^{T′}(γ, f).

The inner sum is over the regular elements of T′_{F} modulo _{0}Z′_{F}. For the T′ occurring in this sum

I_{F} ≠ F^{×}ANm I_{L}.

Since A = ∏ A_{v} this is possible only if A ⊆ Nm I_{L} and then the formula given below shows that μ, which seems to depend on T, is in fact 1. But we prefer not to make use of this until it is necessary, and perhaps not at all.

The missing terms in (5.10) will be extracted from (5.4). It is convenient to perform a sequence of modifications of (5.4) and (5.10) before comparing them. We write

meas (_{0}Z′T′_{F} \T′) = meas (_{0}Z′Z′_{F} \Z′) meas (Z′T′_{F} \T′).

Since meas (_{0}Z′Z′_{F} \Z′) is common to (5.4) and (5.10) it will be ignored. Suppose B = AI_{F^{2}} and let

G′′ = {g | det g ∈ B}.

The various objects associated to G′′ will be denoted in the same way as those associated to G′, except that the prime will be doubled. Since A′′\G′′ = Z′\G′ we may take the measures on the two spaces to be the same and replace the space of integration in (5.4) by Z′′N(A)\G′′. The sum over x appearing there may be replaced by [B_{F} : A_{F} F^{X^{2}}] times a sum over B_{F} \F^{×}.

We want to replace G′ by G′′ and T′ by T′′ in (5.10). Since T′′\G′′ = T′\G′ and Z′′\T′′ = Z′\T′ this is certainly possible. However

meas (Z′′T′′_{F}\T′′) = [T′′_{F} : T′_{F} F^{×^{2}}] meas (Z′′T′′_{F}\T′′)

and if T′ corresponds to the quadratic extension L,

μ = μ(T′) = [F^{×} ∩ A Nm I_{L} : A_{F} Nm L^{×}]

while

μ(T′′) = [F^{×} ∩ B Nm I_{L} : B_{F} Nm L^{×}].

However

A Nm I_{L} = B Nm I_{L};

so that

μ(T′)[T′′_{F} : T′_{F} F^{×^{2}}] = μ(T′′)[B_{F} Nm L^{×} : A_{F} NmL^{×}][B_{F} ∩ NmL^{×} : (A_{F} ∩ NmL^{×}F^{×^{2}}].

The middle factor equals

[B_{F} : B_{F} ∩ A_{F} Nm L^{×}].

Moreover, the map

B_{F} ∩ Nm L^{×} → B_{F} ∩ A_{F} Nm L^{×}/A_{F} F^{×^{2}}

is surjective with kernel

A_{F} F^{×^{2}} ∩ Nm L^{×} = (A_{F} ∩ Nm L^{×})F^{×^{2}}.

Thus the product of the last two factors is [B_{F} : A_{F} F^{×^{2}}]. If we disregard this integer, which is common to (5.4) and (5.10), we may replace G′ by G′′. In order not to burden the notation we now assume that A contains I_{F^{2}} and that G′ is G′′.

The symbols G′′ and B are now free again and we set B = AF^{×} and define G′′ in terms of B as before. The quotient Z′\G′ is open in Z′′\G′′ and we choose the invariant measure on Z′′\G′′ so that it restricts to that on Z′\G′. We may drop the sum over x from (5.4) provided we now integrate over Z′′N(A)\G′′. The space T′\G′ is the same as T′′\T′′G′, which is open in T′′\G′′, and we choose the measures to be compatible. Then 𝔇(T′′/F) = {1} and

∑_{𝔇(T′/F)} Φ^{δ}(γ, f)

is equal to

∫_{T′′\G′′} f(g^{−1}γg)dg.

We now have no choice for the measure on Z′′\T′′. It must be compatible with that on the open subset Z′\T′.

Since

[T′′_{F} ∩ T′ : T′_{F}] = [A_{F} F^{×} ∩ A : A_{F}] = 1,

we have

meas (Z′′T′′_{F}\T′′) = [T′′ : T′′_{F}T′] meas (Z′T′_{F} \T′).

Moreover

[T′′ : T′′_{F}T′] = [AF^{×} ∩ Nm I_{L} : (AF^{×} ∩ Nm L^{×})(A ∩ Nm I_{L})]

which may be simplified to

[AF^{×} ∩ Nm I_{L} : Nm L^{×}(A ∩ Nm I_{L})].

We claim that this index equals

μ(T′) = [F^{×} ∩ A Nm I_{L} : A_{F} Nm L^{×}].

Suppose u in Nm I_{L} equals xy with x ∈ A, y ∈ F^{×}. Then y lies in F^{×} ∩ A Nm I_{L}. If u = vw with v ∈ Nm L^{×}, w ∈ A ∩ Nm I_{L}, then y = (x^{−1}w)v and x^{−1}w = v^{−1}y. Clearly x^{−1}w lies in A_{F}. Conversely, if y ∈ F^{×} ∩ A Nm I_{L} then we may find u ∈ Nm I_{L}, x ∈ A with u = xy. If y = zv with z ∈ A_{F} , v ∈ Nm L^{×} then u = v(xz) and v ∈ Nm L^{×}, xz ∈ A ∩ Nm I_{L}. The conclusion is that we may suppose that A contains both F^{×} and I_{F^{2}}.

The quotient G′\G̃(A) is now compact and abelian. We take its measure to be 1 and write the integral of (5.4) as a sum over its characters

(5.11)  ∑_{κ} ∫_{I_{F} N(A)\_{G}(A)} κ(det g)f (g^{−1}a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) β(g)^{−s}dg.

It should perhaps be stessed that for a given f only finitely many terms of this sum are not 0. By global class field theory each non-trivial κ occurring in this sum determines a quadratic extension of F and the quadratic extensions so obtained are precisely those for which

[𝔈(T′/A) : Im 𝔈(T′/F)] = 2

when T′ is the corresponding Cartan subgroup of G′. Since Z′_{F} is just the set of singular elements in T′_{F} and since there is in (5.4) a sum over_{0}Z′_{F} \Z′_{F}, we can hope that the term of (5.11) corresponding to κ is just the missing term in (5.10) corresponding to a and the T′ defined by κ. The expression

\frac{1}{4} meas (Z′T′_{F} \T′) ∑_{𝔇(T′/A)} κ′(δ)Φ^{δ}(γ, f) = \frac{1}{4} meas (Z′T′_{F} \T′)Φ^{T′}(γ, f)

may be written

\frac{1}{2} meas (I_{F} T̃(F)\T̃(A)) ∫_{_{T}(A)\_{G}(A)} f(g^{−1}γg)κ(det g)dg

if we so normalize measures that

meas (T′\T̃(A)) = 1.

We take κ to be non-trivial and write the corresponding integral in (5.11) as a product

(5.12)  ∏_{v} ∫_{F_{v}^{×}N(F_{v})\_{G}(F_{v})} κ_{v}(det g)f_{v} (g^{−1}a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) β(g)^{−s}dg.

Suppose κ_{v} is unramified, and f_{v} is supported on _{0}Z′_{v}K′_{v} and satisfies

f(zk) = χ(z)f(k).

Then the local integral is equal to

meas (F_{v}^{×}N(F_{v}) ∩ G̃(O_{F_{v}})\G̃(O_{F_{v}})f(a) ∑_{n=0}^{∞} κ_{v}(ϖ_{v}^{n})|ϖ_{v}^{n}|^{1+s}

and the sum is equal to L(1 + s, κ_{v}). Since the global L-function

Π_{v}L(1 + s, κ_{v}) = L(1 + s, κ)

is regular at s = 0 the constant term of the Laurent expansion of (5.12) at s = 0 is

L(1, κ) ∏_{v} \frac{1}{L(1, κ_{v})} ∫_{F_{v}^{×}N(F_{v})\_{G}(F_{v})} κ_{v}(det g)f_{v} (g^{−1}a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg.

We have fixed the measure on N(A) but not on T̃(A) or on I_{F} \T̃(A). Since the results are independent of this choice we may pick any measure that is convenient for the completion of the computations. If Z̃ is the centre of G̃ then I_{F} \T̃(A) is Z̃\T̃(A) and we take the unnormalized Tamagawa measure ([9], §6). The measure on N(A) may also be taken to be the unnormalized Tamagawa measure for it has the desired property that

meas (N(F)\N(A)) = 1.

The lattice of characters of Z̃\T̃ is Z with the action of 𝔊(L/F) = {1, σ} given by σz = −z. With this action

[H^{1}(𝔊(L/F), Z)] = 2.

Moreover, the kernel of

H^{1}(F, Z̃\T̃) → Π_{v}H^{1}(F_{v}, Z̃\T̃)

is trivial. Thus by a general theorem of Ono on the Tamagawa number of a torus

\frac{1}{2} meas (I_{F} T̃(F)\T̃(A)) = L(1, κ).

Finally

∫_{_{T}(A)\_{G}(A)} f(g^{−1}γg)κ(det g)dg = ∏_{v} ∫_{_{T}(F_{v})\_{G}(F_{v})} f_{v}(g^{−1}γg)κ_{v}(det g)dg.

We have seen in §2 that if a ∈ Z′_{v} then

lim_{γ→a} ∫_{_{T}(F_{v})\_{G}(F_{v})} f_{v}(g^{−1}γg)κ_{v}(det g)dg

is equal to a constant c_{v} times

\frac{1}{|a|_{v}L(1, κ_{v})} ∫_{F_{v}^{×}N(F_{v})\_{G}(F_{v})} κ_{v}(det g)f_{v} (g^{−1}a ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ) g) dg.

All we need do is show that c_{v} is 1 for almost all v and that

∏_{v} c_{v} = 1.

We define the local Tamagawa measure on N by means of the form dx if

n = ( \begin{matrix}1 & x \\ 0 & 1 \end{matrix} )

and on Z̃\T̃ as the quotient of

\frac{1}{γ_{1}^{0} − γ_{2}^{0}} \frac{dγ_{1}}{γ_{1}} \frac{dγ_{2}}{γ_{2}}

by

\frac{dz}{z} .

However, we might as well use a more natural Tamagawa measure locally, that obtained by suppressing the factor L(1, λ_{v}) from the definition in [9]. We must then suppress the factor L(1, κ_{v}) in the definition of c_{v} as well. We may assume that the form η on

A = \{( \begin{matrix}α & 0 \\ 0 & β \end{matrix} )\}

appearing in Lemma 6.1 of [9] is

\frac{dα}{α} \frac{dβ}{β}

and that the measure on Z̃ is given by the form dz/z. Then Lemma 2.1 together with Definitions 2.2 and 2.3 and the discussion following the aforementioned Lemma 6.1 show that

c_{v} = |(γ_{1}^{0} − γ_{2}^{0})^{2}|^{1/2}.

We have still to consider the contribution to (5.4) from the trivial character as well as the remaining terms of the trace formula, but we first examine the contribution

\frac{1}{4} ∑_{T′} μ(T′) meas (_{0}Z′T′_{F} \T′)Σ ϕ^{T′}(γ, f)

more closely. Applying the trace formula to the pairs T′_{F} , T′ we see that this equals

\frac{1}{4} ∑_{T′} μ(T′) ∑_{θ}⟨θ, Φ^{T′}(f)⟩

where θ runs through all characters of T′_{F} \T′ which equal κχ^{−1} on_{0}Z′. Appealing to the discussion of §2 we transform this to

\frac{1}{4} ∑_{T′} μ(T′) ∑_{θ} Π_{v}(∑_{π_{v}∈Π^{+}(θ_{v})} χ_{π_{v}}(f_{v}) − ∑_{π_{v}∈Π^{−}(θ_{v})} χ_{π_{v}}(f_{v})).

It should perhaps be stressed that for almost all v the trace χ_{π_{v}}(f_{v}) is 0 when π_{v} ∈ Π^{−}(θ_{v}). Moreover, when v splits, Π^{−}(θ_{v}) is empty and Π^{+}(θ_{v}) consists of the components of the principal series defined by θ_{v}.

We distinguish three types of θ. Let γ → γ̄ be the automorphism of T′ or of T̃(A) corresponding to conjugation on the corresponding quadratic field. We first introduce a provisional classification.

a_{1}) On T′

θ(γ/γ̄) ≢ 1.
b_{1}) On T′
θ(γ/γ̄) ≡ 1

but θ cannot be extended to a character θ̃ of T̃(F)\T̃(A) satisfying

θ̃(γ) ≡ θ̃(γ̄).

c_{1}) θ can be extended to a character of T̃(F)\T̃(A) satisfying this identity.

Suppose there are two elliptic Cartan subgroups T′_{1}, T′_{2} which are not stably conjugate and two characters θ^{1} and θ^{2} for which

Π(θ^{i}) = {⊗ π_{v} | π_{v} ∈ Π(θ_{v}^{i}) = Π^{+}(θ_{v}^{i}) ∪ Π^{−}(θ_{v}^{i})}

are the same. It is understood that in each product π_{v} contains the trivial representation of K′_{v} for almost all v. Suppose also that

(5.13)  [𝔈(T′_{i}/A) : Im 𝔈(T′_{i}/F)] = 2.

Extend θ^{i} to a character θ̃^{i} of T̃(F)\T̃(A). Then π(θ̃_{v}^{1}) and π(θ̃_{v}^{2}) determine the same Lindistinguishable class of G(F_{v}) for all v. We may regard θ̃^{i} as a character of L_{i}^{×}\I_{L_{i}} and consider

ρ^{i} = Ind (W_{L_{i}/F} , W_{L_{i}/L_{i}}, θ̃^{i}).

L-indistinguishability 43 When we compose ρ^{i} with

GL(2, C) → P GL(2, C) → GL(3, C)

we obtain three-dimensional representations which are locally equivalent everywhere and hence, by Lemma 12.3 of [6], equivalent. It is easy to see that this is possible only if

γ → θ̃^{i}(γ/γ̄)

is of order 2 but not trivial. In addition F^{×}NmI_{L_{2}} must contain

F^{×}{Nm_{L_{1}/F} x|θ^{1}(x/x̄) = 1}.

Indeed this set must be

F^{×}NmI_{L_{1}} ∩ F^{×}NmI_{L_{2}}.

It follows from (5.13) that T_{1}, θ^{1} must be of type (b_{1}).

Conversely if the pair (T_{1}, θ^{1}) is of type (b_{1}) and θ^{1} lifts to θ̃^{1} then

γ → θ^{1}(γ/γ̄)

is of order 2. There are exactly three different quadratic extensions L_{1}, L_{2}, L_{3} of F for which

F^{×}NmI_{L_{i}} ⊇ F^{×}{Nm_{L_{1}/F} x|θ^{1}(x/x̄) = 1}.

There are also three characters θ̃^{1}, θ̃^{2}, and θ̃^{3} such that the representations

ρ^{i} = Ind (W_{L_{i}/F} , W_{L_{i}/L_{i}}, θ̃^{i})

become equivalent upon inflation. Thus

Π(θ^{1}) = Π(θ^{2}) = Π(θ^{3}),

if θ^{i} is the restriction of θ̃^{i} to T^{i}. We say that (T_{1}, θ^{1}) is of type (b′_{1}) if

A ⊆ ∩_{i} F^{×}NmI_{L_{i}}

and of type (b′′_{1}) otherwise.

The final classification is:

type (a) ⇔ type (a_{1}) or type (b′′_{1});
type (b) ⇔ type (b′_{1});
type (c) ⇔ type (c_{1}).

If θ is of type (c) and extends to θ̃ with θ̃(γ) ≡ θ̃(γ̄) then

θ̃(γ) = ω(Nm γ)

and the elements of Π(θ) are the components of the principal series associated to the character

η : ( \begin{matrix}α & 0 \\ 0 & β \end{matrix} ) → ω(αβ)κ(β)

of A′. Here κ is the quadratic character associated to the quadratic extension defined by T′. We do not expect to see them in the discrete spectrum or in the stable or labile part of the trace formula, and it turns out that the corresponding contribution to (5.14) is cancelled by the labile part of (5.5).

The operator M(η) appearing in (5.5) is equal to

\frac{L(1, μ)}{L(1, μ^{−1})} ⊗_{v} R(η_{v}).

Here μ is of order 2. If it is trivial then the quotient L(1, μ)/L(1, μ^{−1}), which is defined as a limit, is equal to −1. Since each R(η_{v}) is then the identity, the corresponding contribution to the trace is stable. If μ is not trivial, the quotient is 1 and by Lemmas 3.5 and 3.6

∏_{v} trace R(η_{v})ρ(f, η_{v})

is equal to

∏_{v}(∑_{π_{v}∈Π^{+}(θ_{v})} χ_{π_{v}}(f_{v}) − ∑_{π_{v}∈Π^{−}(θ_{v})} χ_{π_{v}}(f_{v})).

Observe that class field theory associates to μ a quadratic extension L. T′ is the corresponding Cartan subgroup and θ is defined by

θ(x) = η (( \begin{matrix}Nm x & 0 \\ 0 & 1 \end{matrix} )) .

For α in A,

μ(α) = η (( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} )) η_{1} (( \begin{matrix}α^{−1} & 0 \\ 0 & 1 \end{matrix} )) = 1

because η = η_{1}. Thus A ⊆ F^{×}NmI_{L} and

[𝔈(T′/A) : Im 𝔊(T′/F)] = 2.

In order to verify that the expected cancellation occurs we have to check that if θ extends to a character θ̃ of T̃(F)\T̃(A) with θ̃(x) ≡ θ̃(x̄) then there are exactly μ(T′) characters η with η = η_{1} yielding the pair (T, θ). If α ∈ Nm I_{L} ∩ A we must have

η (( \begin{matrix}α & 0 \\ 0 & 1 \end{matrix} )) = θ(β), α = Nm β,

and we must have

η (( \begin{matrix}α & 0 \\ 0 & α^{−1} \end{matrix} )) = κ(α)

if κ is the quadratic character associated to T′. Then

η(z) = κ(z)θ(z), z ∈ Z′.

In particular η is equal to χ^{−1} on _{0}Z′. The number of possibilities for η is therefore

[A : A_{F} (Nm I_{L} ∩ A)].

However, A is contained in F^{×}NmI_{L} and if we write a = xy then a → x yields an injection of A/A_{F} (NmI_{L} ∩ A) into F^{×}/A_{F} NmL^{×}. The image is

F^{×} ∩ ANmI_{L}/A_{F} NmL^{×}

and the order of this group is μ(T′).

The terms (5.1) and (5.6) are clearly stably invariant. We shall show that the last term of the trace formula combines with (5.3) and the remaining part of (5.4) to give a stably invariant distribution. Denote these three terms by S_{1}, S_{2}, S_{3}. We take g = ∏ g_{v} in G̃(A) and show that

∑_{i=1}^{3}(S_{i}^{g} − S_{i}) = 0.

According to Lemma 3.4 the distribution S_{1}^{g^{−1}} − S_{1} is given by

∑_{v} \frac{1}{2π} ∫_{D^{0}}\{trace ρ(f_{v}, η_{v})N(g_{v})\} \{ ∏_{w≠v} trace ρ(f_{w}, η_{w})\} |ds|.

For brevity denote the function

trace ρ(f_{v}, η_{v})N(g_{v})

by H_{v}(η_{v}) and the function trace ρ(f_{w}, η_{w}) by I_{w}(η_{w}). Then

(5.15)  H_{v}(η_{v}) ∏_{w} I_{w}(η_{w})

is a function on DA^{0}, the set of all characters of A′ which are of absolute value 1 and equal χ^{−1} on 0Z′. According to Lemma 3.2 the difference S_{2}^{g}(f) − S_{2}(f) is equal to

−λ ∑_{v} ∑_{_{0}Z′_{F} \A_{F′}} \frac{Hv^{∨}(γ)}{L(1, f_{v})} ∏_{w≠v} \frac{Iw^{∨}(γ)}{L(1, F_{w})}.

Here Hv^{∨} and Iw^{∨} are the local Fourier transforms of H_{v} and I_{w}. The global Fourier transform of (5.15) is

γ → λ\frac{Hv^{∨}(γ_{v})}{L(1, F_{v})} ∏_{w≠v} \frac{Iw^{∨}(γ_{w})}{L(1, F_{w})}.

If we can show that S_{3}^{g}(f) − S_{3}(f) is equal to

−λ ∑_{v} ∑_{_{0}Z_{F′} \Z_{F′}} \frac{Hv^{∨}(γ)}{L(1, F_{v})} ∏_{w=v} \frac{Iw^{∨}(γ)}{L(1, F_{w})}

when Z′_{F} is the group of scalar matrices in G′, then an appeal to the Poisson summation formula will establish that the sum of the three differences is 0.

The value of S_{3} at f is the constant term of the Laurent expansion at 0 of

∑_{_{0}Z_{F′} \Z_{F′}} L(1 + s, 1_{F}) ∏_{v} θ(a, s, f_{v}).

The distribution θ(a, 0, f_{v}) is stably invariant. Hence

S_{3}(f^{g^{−1}}) − S_{3}(f)

is equal to

λ ∑_{_{0}Z_{F′} \Z_{F′}} ∑_{v}{θ′(a, 0, f_{v}^{g^{−1}}) − θ′(a, 0, f_{v})} {∏_{w} θ(a, 0, f_{w})}

if λ is the residue of L(1 + s, 1_{F}) at s = 0. We now invoke Lemma 3.3.

6. Consequences. Let m(π) be the multiplicity with which an irreducible admissible representation of G′ occurs in the representation r. Then

trace r(f) = ∑ m(π) trace π(f).

If π belongs to no Π(θ) with

[𝔈(T′/A) : Im𝔈(T′/F)] = 2

we set n(π) = m(π). If π belongs to some such Π(θ) we say that π is of type (a) or type (b) according as θ is of type (a) or (b).

If π is of type (a) and θ^{1} is a character of T′_{1} then π can belong to Π(θ^{1}) only if T′_{1} and T′ are stably conjugate. If T′_{1} = T′ then

Π^{+}(θ_{v}) = Π^{+}(θ_{v}^{1}) Π^{−}(θ_{v}) = Π^{−}(θ_{v}^{1})

for all v. Let e(π) be the number of characters of T′_{F} \T′ for which π ∈ Π(θ). Let ⟨ϵ, π_{v}⟩ be 1 or −1 according as π_{v} ∈ Π^{+}(θ_{v}) or π_{v} ∈ Π^{−}(θ_{v}). Then ⟨ϵ, π_{v}⟩ is 1 for almost all v, and we set

⟨ϵ, π⟩ = Π_{v}⟨ϵ, π_{v}⟩.

The local factors depend upon a number of choices but ⟨ϵ, π⟩ itself is well-defined when the local choices are made to depend in a consistent manner on global data, and this was done. Finally set

n(π) = m(π) − \frac{1}{4}⟨ϵ, π⟩e(π)μ(T′).

If π is of type (c) there are three distinct quadratic extensions L_{1}, L_{2}, L_{3} with associated Cartan subgroups T′_{i} satisfying

[𝔈(T′_{i}/A), Im 𝔈(T′_{i}/F)] = 2

and characters θ^{i} of T′_{i} with π ∈ Π(θ^{i}). Let e_{i}(π) be the number of characters of T′_{i} trivial on T′_{i} ∩ G′_{F} for which π ∈ Π(θ^{i}). For each i we may introduce ⟨ϵ_{i}, π_{v}⟩ and set

⟨ϵ_{i}, π⟩ = Π_{v}⟨ϵ_{i}, π_{v}⟩.

Then we introduce

n(π) = m(π) − \frac{1}{4} ∑_{i=1}^{3}⟨ϵ_{i}, π⟩e_{i}(π)μ(T′_{i}).

There is a curious property of the triple ⟨ϵ_{1}, π⟩, ⟨ϵ_{2}, π⟩, ⟨ϵ_{3}, π⟩ which should be remarked. If we let L_{i} = F (τ_{i}) then, as at the end of §2, τ_{i} determines a particular T′_{i} and a particular γ_{i}^{0} ∈ T_{i}(F̃). These we use to define Π^{+}(θ_{v}^{i}) and Π^{−}(θ_{v}^{i}). The set Π(θ_{v}^{i}) is the same for all i and a π_{v} in the set acts on the space of functions in the Kirillov model supported by

{a ∈ F_{v}^{×}|κ_{v}^{i}(a) = ⟨ϵ_{i}, π_{v}⟩, i = 1, 2, 3}.

Consequently

⟨ϵ_{1}, π_{v}⟩⟨ϵ_{2}, π_{v}⟩ = ⟨ϵ_{3}, π_{v}⟩

and

⟨ϵ_{1}, π⟩ ⟨ϵ_{2}, π⟩ = ⟨ϵ_{3}, π⟩.

The distribution

f → ∑_{π} n(π) trace π(f)

is stable.

Lemma 6.1. If g ∈ G̃(A) define π^{g} by

π^{g} : h → π(ghg^{−1}).

Then n(π^{g}) = n(π).

Set

l(π) = n(π^{g}) − n(π)

and let B be the space of finite linear combinations of functions to which we have applied the trace formula. B is closed under f → f^{∗} with

f^{∗}(g) = f̄(g^{−1})

and under the obvious convolution product. Moreover, for f in B

∑ l(π) trace π(f)

is absolutely convergent and equal to 0. In particular

(6.1)  ∑_{l(π)>0} l(π) trace π(ff^{∗}) = ∑_{l(π)<0} −l(π) trace π(ff∗).

Let X^{+} be the set of π with l(π) > 0 and X^{−} the set with l(π) < 0. All we need do is show that X^{−} is empty. If not, choose f so that the maximum eigenvalue λ(π) of π(ff^{∗}), π ∈ X^{−}, is 1. Let λ(π_{0}) = 1 with π_{0} in X^{−} and let x_{0} be a unit vector in the space of π_{0} with π_{0}(ff^{∗})x_{0} = x_{0}.

Set

δ = max_{π∈X^{+}}λ(π).

δ is finite because the representations π are unitary. Moreover, it is positive. If f_{1} lies in B let ∥π(f_{1})∥ be the Hilbert-Schmidt norm of π(f_{1}). As on p. 498 of [6] we may choose f_{1} so that

∑_{π∈X^{+}} l(π)∥π(f_{1})∥^{2} < −l(π_{0})/2δ

and so that

∥π_{0}(f_{1})x_{0}∥ = ∥x_{0}∥.

In (6.1) we replace f by f_{1}f. The left side is then less than −l(π_{0})/2. The right side is at least

−l(π_{0}) trace π_{0}(f_{1}ff^{∗}f_{1}^{∗}) = −l(π_{0}) trace π_{0}(f_{1}^{∗}f_{1}ff^{∗})

and this is greater than or equal to

−l(π_{0})∥π_{0}(f_{1})x_{0}∥^{2} = −l(π_{0}).

The resulting contradiction proves the lemma.

Suppose π̃ = ⊗ π̃_{v} is a representation of G̃(A) and let π be one of the irreducible components of its restriction to G(A). Let G(π) or G(π̃) be the set of all h in G̃(A) for which^{h}π : g → π(h^{−1}gh) is equivalent to π. Let X(π) = X(π̃) be the set of all characters of ω of I_{F} for which π̃ ≃ ω ⊗ π̃. X(π̃) consists of the characters trivial on {det h|h ∈ G(π̃)}. Let Y be the set of characters of F^{×}I_{F} and Y (π̃) the set of all characters ω of I_{F} for which ω ⊗ π̃ is automorphic and cuspidal. Let

q(π) = q(π̃) = [Y (π̃)/Y X(π̃)].

It will be seen in a moment that this index is finite; it is likely always to be 0 or 1 and to be 0 if π̃ = π(μ, ν) with two characters μ, ν in Y; but the only proof we can envisage at the moment is difficult and lies beyond the scope of this paper.

Lemma 6.2. Suppose G′ = G(A) and π is infinite-dimensional with π(z) = χ^{−1}(z)I, z ∈ _{0}Z′. Then

q(π) = ∑_{_{G}(F)G(π)\_{G}(A)} m(^{h}π).

It follows easily from this equation that q(π′) is finite. We may suppose that _{0}Z′ = Z′. If Z̃ is the centre of G̃ let G′′ = Z̃(A)G′ = Z′′G′. From the relation

F^{×} ∩ I_{F}^{2} = F^{×^{2}},

we conclude that

G′ ∩ Z′′G′′_{F} = Z′G′_{F}

and that

G′_{F} Z′\G′ ≃ G′′_{F} Z′′\G′′.

Thus if we extend χ to a character χ′′ of Z′′ and let π′′ be the corresponding extension of π to G′′, we need only show that

q(π) = ∑_{_{G}(F)G(π)\_{G}(A)} m(^{h}π′′).

Let L′′, L_{1}, and L̃ be the spaces of cusp forms on G′′_{F} \G′′, G̃(F)\G′′G̃(F), and G̃(F)\G̃(A) transforming according to a given character of Z′′, and let s′′, s_{1}, and s̃ be the representations of the three groups G′′, G′′G̃(F), and G̃(A) on these three spaces. L′′ and L are the same as spaces. Suppose π′′ is a representation occurring in s′′ with multiplicity m(π′′). Let π be its restriction to G′ and let

G_{1}(π) = G′′G̃(F) ∩ G(π).

It follows easily from Lemma 2.6 that π′′ extends to a representation σ of G(π) on the same space. Let σ_{1} be the restriction of σ to G_{1}(π).

The subspace V′′ of L′′ transforming according to π′′ transforms under G_{1}(π) according to

⊕_{i=1}^{m(π′′)}ω_{i} ⊗ σ_{1}

where ω_{i} is a character of G′′\G_{1}(π). The smallest invariant subspace of L_{1} containing V′ transforms according to

⊕_{i} Ind (G′′G̃(F), G_{1}(π), ω_{i} ⊗ σ_{1}),

and each summand is irreducible.

Since

s̃ = Ind (G̃(A), G′′G̃(F), s_{1})

we have

s̃ = ⊕_{{π}} ⊕_{i} Ind (G̃(A), G_{1}(π), ω_{i} ⊗ σ_{1}).

In this sum two representations π and π_{1} are taken to be equivalent if π_{1} = ^{h}π with det h ∈ F^{×}. The induction can be carried out in two steps, first from G_{1}(π) to G(π), and then from G(π) to G̃(A). The quotient G_{1}(π)\G(π) is a subquotient of F^{×}I_{F}^{2} \I_{F} and hence compact; and

Ind (G(π), G_{1}(π), ω_{i} ⊗ σ_{1}) = ⊕_{ω}ω ⊗ σ.

The sum is over all characters ω of G(π) which agree with ω_{i} on G_{1}(π). At the second step we obtain the summands

Ind (G̃(A), G(π), ω ⊗ σ),

which are irreducible. Since each of these representations contains π′′, any one of them arises also from π′′_{1} if and only if π′′_{1} = ^{h}π′′ with h ∈ G̃(A). The lemma is therefore clear if q(π) = 0.

If q(π) ≠ 0 we may take π̃ to be automorphic and cuspidal. Let Y′(π) and Y′ be the elements of order two in Y (π) and Y. All elements of X(π) are of order two and every element of Y (π) is trivial on F^{×^{2}}. Since

F^{×}I_{F}^{2} /F^{×^{2}} ≃ F^{×}/F^{×^{2}} × I_{F}^{2} /F^{×^{2}}

the set Y (π) is equal to Y′(π)Y and

q(π) = [Y′(π)/Y′X(π)].

The lemma now follows easily from multiplicity one for G̃(A) and the observation that a character lies in Y′X(π) if and only if it is trivial on G_{1}(π).

Now suppose G′ is arbitrary and π′ a cuspidal automorphic representation. We attempt to compute m(π′) in terms of the m(π) for cuspidal automorphic forms for G(A). We may suppose that _{0}Z′ = Z′. We may also replace G(A) by G′′ = Z′G(A) because

G(A) ∩ Z′G′′_{F} = Z(A)G(F)

and

Z(A)G(F)\G(A) = Z′G′′_{F} \G′′.

We proceed as in the proof of the lemma with G′ replacing G̃(A). Set

G′(π) = G′ ∩ G(π), G′_{1}(π) = G′′G′_{F} ∩ G(π).

Let σ′ and σ′_{1} be the restriction of σ and σ_{1} to G′(π) and G′_{1}(π). We obtain a direct sum decomposition of the representation of G′ on the space of cusp forms transforming according to χ^{−1} under Z′. It is

⊕_{{π′′}} ⊕_{i=1}^{m(π′′)} ⊕_{ω′} Ind (G′, G′(π), ω′ ⊗ σ′).

For the purposes of this sum π′′ and π′′_{1} are taken as equivalent if

π′′_{1} ≃ ^{h}π′′, det h ∈ A_{F} .

For each π′′ and each i there is a character ω′_{i} of G′′\G′_{1}(π) and the inner sum is over all characters ω′ of G′(π) whose restriction to G′_{1}(π) is ω′_{i}. The space of functions on

G′′_{F} \G′′ = G′_{F} \G′′G′_{F}

transforming according to π′′ transforms under G′_{1}(π) according to

⊕_{i=1}^{m(π′′)}ω_{i} ⊗ σ′_{1}.

If the summands corresponding to π′′, i, ω′, and π̃′′, j, ω̄′ are equivalent then

(6.2)  π̃′′ ≃ ^{h}π′′, h ∈ G′.

Suppose that (6.2) is satisfied and that q(π) is 1. Then in addition π̃′′ ≃ ^{g}π′′ with g ∈ G̃(F). Thus h^{−1}g ∈ G(π). Also m(π′′) = m(π̃′′) = 1, and

ω̄′_{1} ⊗ σ̄′_{1} ≃ ^{g}(ω′_{1} ⊗ σ′_{1}).

Thus

ω̄′_{1} ⊗ σ̄′_{1} ≃ ^{h}(ω′_{1} ⊗ σ′_{1})

and the summand indexed by π′′, 1 and any ω′ is equivalent to that indexed by π̃′′, 1, and some ω′. We have established:

Corollary 6.3. Suppose π′ is a cuspidal automorphic representation of G′ and π one of the irreducible components of the restriction of π′ to G(A). Let A(π′) = A(π) = {det g|g ∈ G(π)}. If q(π) is 1 then the multiplicity with which π′ occurs in the space of automorphic forms on G′_{F} \G′ is

[AA(π) ∩ F^{×}A(π) : A_{F} A(π))] = [F^{×} ∩ AA(π) : A_{F} A(π)_{F}].

Proposition 6.4. Suppose π is a representation of G′. If m(π^{g}) is not equal to m(π) for all g ∈ G̃(A) there is a T′ with [𝔈(T′/A) : Im 𝔈(T′/F)] = 2 and a θ such that π belongs to Π(θ).

If there were no such T′ and θ then m(π^{g}) would be n(π^{g}) for all g.

Proposition 6.5. Suppose ω is a non-trivial character of F^{×}\I_{F} of order 2 and π is a constituent of the space of cusp forms of G̃(A). If π ≃ ω ⊗ π then there is a character θ of L^{×}\I_{L}, where L is the quadratic extension of F defined by ω, for which π = π(θ).

Let G′ be the group defined by

A = Nm_{L/F} I_{L}.

The restriction of π to G̃(F)G′ is the direct sum of two irreducible representations π and π_{1}. Let π′ and π′_{1} be irreducible components of the restrictions of π and π_{1} to G′. One of π′ and π′_{1} must occur in the space of cusp forms on G′. Suppose they both do. Taking the Fourier expansion with respect to the group N(F)\N(A) we see that there are two characters ψ and ψ_{1} of F \A such that π′ is contained in Ind (G′, N(A), ψ) and π′_{1} in Ind (G′, N(A), ψ_{1}). Let

ψ(x) = ψ_{1}(βx) β ∈ F^{×}.

There is a g in G̃(A) with ω(α) = −1 if α = det g for which π is equivalent to

h → π_{1}(ghg^{−1}).

Thus π is also contained in Ind (G′, N(A), ψ′) if ψ′(x) = ψ_{1}(αx). Since G′_{v} = G(π′_{v}) for all v it follows from Corollary 2.7 that ω(β) = ω(α). This is a contradiction, and so one of m(π′) and m(π′_{1}) is 0. The proof is completed by an appeal to Proposition 6.4.

Corollary 6.6. Suppose L is a quadratic extension of F, θ a character of L^{×}\I_{L} which does not factor through the norm, and π̃ = π(θ). Then q(π̃) = 1.

If ω is the quadratic character of F^{×}\I_{F} defined by L then

ω ⊗ π̃ ≃ π̃.

Suppose ν is a character of I_{F} and π̃′ = ν ⊗π̃ is also a constituent of the space of cusp forms. Since ω ⊗ π̃′ ≃ π̃′ there is a θ′ for which π̃′ = π(θ′). The two characters θ and θ′ define representations of the Weil group W_{L/F} in GL(2, C) and because π̃′ = ν ⊗ π̃ the three-dimensional representations obtained from

GL(2, C) → P GL(2, C) → GL(3, C)

have equivalent restrictions to the local Weil groups at every place. Glancing at Lemma 12.3 of [6], we conclude that the three-dimensional representations of W_{L/F} itself are equivalent. Writing the representations out explicitly and recalling that the first cohomology group of L^{×}\I_{L} is trivial, we see that

θ′(x) = ω′(Nm x)θ

with some character of F^{×}\I_{F}. Thus

π̃′ ∼ ω′ ⊗ π̃.

We suppose more generally that π̃ = π(ρ) where ρ is an irreducible two-dimensional representation of the Weil group. We assume that for every place v and every character ω_{v} of F_{v}^{×},

π(ρ_{v}) ≃ ω_{v} ⊗ π(ρ_{v})

if and only if

ρ_{v} ≃ ω_{v} ⊗ ρ_{v}.

This is known in general, and in the case of primary concern to us that ρ is induced it is a consequence of the discussion of §2.

There are two notions of equivalence on the set of representations ρ′ = ω ⊗ ρ, ω a character of F^{×}\I_{F}.

i) Global: ρ′ ∼ ρ if and only if the representation ρ′ is equivalent to ω′ ⊗ ρ with ω′ trivial on AF^{×}.

ii) Local: ρ′ ∼ ρ if and only if for every place v the representation ρ′_{v} is equivalent to ω′_{v} ⊗ ρ_{v} with ω′_{v} trivial on A_{v}.

Global equivalence means that

ω = ω′ω′′

with ω′′ trivial on A(π)F^{×}. Such a factorization is possible if and only if ω is trivial on AF^{×} ∩ A(π)F^{×}. Local equivalence means that for every v

ω_{v} = ω′_{v}ω′′_{v}

with ω′′_{v} trivial on A(π)_{v}. This factorization is possible if and only if ω is trivial on F^{×}(A ∩ A(π)). Thus the number of global equivalence classes within one local equivalence class is

[AF^{×} ∩ A(π)F^{×} : F^{×}(A ∩ A(π))] = [A ∩ A(π)F^{×} : A_{F} (A ∩ A(π))].

This is the index of Corollary 6.3. It depends only on the L-indistinguishability class to which π belongs, that is, it is the same for π and for ^{g}π, g ∈ G̃(A). We denote it by d(π). More generally if π′ is a representation of G′ and π a component of its restriction to G(Ã) we set d(π′) = d(π).

Suppose L is a quadratic extension and

ρ_{i} = Ind (W_{L/F} , W_{L/L}, θ_{i}).

The representations π(ρ_{i}) = π(θ_{i}) determine the same L-indistinguishability class of representations of G′ if and only if

ρ_{2} ∼ ω ⊗ ρ_{1}

with ω trivial on F^{×}(A ∩ A(π̃)), π̃ = π(θ_{1}). However, ρ_{2} ∼ ω ⊗ ρ_{1} if and only if

θ_{2}(γ) ≡ ω(Nm γ)θ_{1}(γ)

or

θ_{2}(γ) ≡ ω(Nm γ)θ_{1}(γ̄).

The bar denotes conjugation on L. However, θ_{1} and θ_{2} have the same restrictions to T′ if and only if

θ_{2}(γ) = ω(Nm γ)θ_{1}(γ)

with ω trivial on F^{×}(A ∩ Nm I_{L}).

Thus the number of characters θ′ of T′_{F} \T′ which yield the same L-indistinguishability class Π(θ′) is

2[F^{×}(A ∩ Nm I_{L}) : F^{×}(A ∩ A(π))]

unless

θ_{1}(x/x̄) = 1

when it is simply

[F^{×}{A ∩ Nm I_{L}) : F^{×}(A ∩ A(π))].

We are going to multiply the latter number by μ(T′). Any element a of A ∩ F^{×} Nm I_{L} is of course also equal to xy, x ∈ F^{×}, y ∈ Nm I_{L} and the map a → x yields an isomorphism

A ∩ F^{×} Nm I_{L}/A_{F} (A ∩ Nm I_{L}) ≃ (F^{×} ∩ A Nm I_{L})/A_{F} Nm L^{×}.

Thus

μ(T′) = [F^{×}A ∩ F^{×}Nm I_{L} : F^{×}(A ∩ Nm I_{L})]

and the product is

[F^{×}A ∩ F^{×}Nm I_{L} : F^{×}(A ∩ A(π̃))].

If π′ lies in Π(θ′) this is d(π′) times

(6.3)  [F^{×}A ∩ F^{X}Nm I_{L} : F^{×}A ∩ F^{X}A(π̃)] = [A ∩ F^{×}Nm I_{L} : A ∩ F^{×}A(π̃)].

Take a character θ of L^{×} and let π̃ = π(θ). Every character of I_{F} /A(π̃)F^{×} is of order two and π̃ ∼ ω ⊗ π̃. Thus I_{F} /A(π̃)F^{×} is of order 1, 2, or 4 and is of order 4 if and only if

x → θ(x/x̄)

is of order two but not identically 1. There are then three different quadratic extensions L = L_{1}, L_{2}, L_{3} and characters θ_{1}, θ_{2}, θ_{3} such that π̃ = π(θ_{i}). In addition

A(π̃)F^{×} = ∩_{i=1}^{3} F^{×} Nm I_{L_{i}}.

If A ⊆ F^{×} Nm L then the index (6.3) is 1 unless

[I_{F} : A(π̃)F^{×}] = 4

and

A ⊈ F^{×} Nm I_{L_{2}} A ⊈ F^{×} Nm I_{L_{3}}

when it is 2.

Suppose θ does not factor through the norm. Since q(π̃) is 1 the discussion culminating in Corollary 6.3 shows in addition that for π′ in the L-indistinguishable class Π(θ′) defined by π̃, m(π′) is 0 or d(π′), that for one of these π′,

m(π′) = d(π′),

and that then

m(^{g}π′) = d(π′)

if and only if det g ∈ AA(π̃)F^{×}. The results of the previous paragraph yield in combination with the considerations above a more precise statement.

Proposition 6.7. Suppose θ is a character of T′_{F} \T′ with

[𝔈(T′/A) : Im 𝔈(T′/F)] = 2

and π in Π(θ) is of type (a). Then

n(π) = d(π)/2

and

m(π) = \frac{d(π)}{2}(1 + ⟨ϵ, π⟩).

One need only observe that when θ is of type (a)

e(π)μ(T′) = 2d(π).

Proposition 6.8. Suppose π is of type (b) and lies in Π(θ^{1}), Π(θ^{2}), Π(θ^{3}), where θ^{i} is a character of T′_{i} trivial on T′_{i} ∩ G′_{F} and

[𝔈(T′_{i}/A) : Im 𝔈(T′_{i}/F)] = 2

Then

n(π) = d(π)/4

and

m(π) = \frac{d(π)}{4}\{1 + ⟨ϵ_{1}, π⟩ + ⟨ϵ_{2}, π⟩ + ⟨ϵ_{3}, π⟩\}.

When θ is of type (b)

e(π)μ(T′) = d(π).

There is a more suggestive way to state the propositions. We first work locally. Suppose F is an extension of E such that

G̃(F) = Res_{F/E}G̃(E)

and the determinant map is from

Res_{F/E}G̃(F) → Res_{F/E}GL(1).

We replace A by A(E) where A is a connected algebraic subgroup of Res_{F/E}GL(1) defined over E. The group G′ is now G(E) where G is the inverse image of A. Let ^{L}G be the associate group of G. If π̃ = π(ρ̃) is the irreducible, admissible representation associated to a two-dimensional representation of the Weil group W_{F} over F then the components of the restriction of π̃ to G′ form the L-indistinguishable class Π(ρ) associated to the corresponding homomorphism ρ of W_{E} to ^{L}G, a quotient of the associate group of Res_{F/E}G̃ by a central torus. Let S be the centralizer of ρ(W_{E}) in ^{L}G^{0} and S^{0} the product of its connected component and its intersection with the centre of ^{L}G^{0}. We shall show that the quotient S^{0}\S is abelian and that the set Π(ρ) may be mapped in a natural and bijective manner to its dual.

We first remind ourselves of the definitions of ρ and ^{L}G [10]. If K is a large Galois extension of E containing F then

X^{∗} = X^{∗}(Res_{F/E}GL(1)) = Ind (𝔊(K/E), 𝔊(K/F), 1) = ⊕_{𝔊(K/F)\𝔊(K/E)}Z

and

X^{∗}(A) = X^{∗}/Y^{∗}.

Let Y_{∗} be the orthogonal complement of Y^{∗} in the dual module. The group ^{L}G is the quotient of the semi-direct product

∏𝔊(K/F)\𝔊(K/E) GL(2, C) × 𝔊(K/F)

by

LZ^{0} = \{∏ ( \begin{matrix}z_{σ} & 0 \\ 0 & z_{σ} \end{matrix} ) | ∏_{𝔊(K/F)\𝔊(K/E)} λ_{σ}(z_{σ}) = 1 for all λ = (λ_{σ}) in Y_{∗}\} .

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

vw = d_{v}(w)v′, d_{v}(w) ∈ W_{K/F} .

If w → σ in 𝔊(K/F) then ρ(w) is the image in ^{L}G of

(∏ ρ̄(d_{v}(w))) × σ.

Notice that the cosets in 𝔊(K/F)\𝔊(K/E) may also be labelled by the v.

We may suppose that one of the v is 1. Suppose ∏ a_{v} commutes with ρ(W_{K/F}). Then

a_{v}ρ̃(d_{v}(w))a_{v′}^{−1} = z_{v}ρ̃(d_{v}(w))

with

∏_{v} λ_{v}(z_{v}) = 1

for all λ = (λ_{v}) in ^{L}Z^{0}. If ρ̃ is a representation by scalar matrices then S is the product of its intersection with the centre and the image of GL(2, C) in ^{L}G^{0} under the diagonal map. Thus S^{0}\S = 1. In this case Π(ρ) consists, as we have seen, of a single element. If ρ̃ is not a representation by scalar matrices and is not induced from a one-dimensional representation of a Weil group over a quadratic extension then the associated projective representation is irreducible and S is contained in the centre of ^{L}G^{0}. Again S^{0}\S = 1 and Π(ρ) consists of a single element.

Suppose

ρ̃ = Ind(W_{K/F} , W_{K/L}, θ̃)

where θ̃ is a one-dimensional representation of W_{K/L}, that is, a character of L^{×}. We suppose θ̃ does not factor through the norm. Suppose T̃ is a Cartan subgroup of G̃ associated to L and T = G ∩ Res_{F/E}T̃. The associate group ^{L}T̃ is a semi-direct product

(C^{×} × C^{×})𝔊(K/F).

𝔊(L/F) acts on C^{×} × C^{×} by permuting the two factors and 𝔊(K/F) acts through its projection on 𝔊(L/F). θ̃ may be regarded as a character of T̃(F) and is associated to a homomorphism φ̃ : W_{K/F} → ^{L}T̃. There is a homomorphism

ψ̃ : ^{L}T̃ → ^{L}G̃

given by

ψ̃ : (a, b) × σ → ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) × σ

if σ acts trivially on L and by

ψ̃ : (a, b) × σ → ( \begin{matrix}a & 0 \\ 0 & b \end{matrix} ) ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) × σ

if σ does not act trivially on L. Then ρ̃ = ψ̃ ∘ φ̃ and π(ρ̃) is the image of θ̃ under the map ψ̃_{∗} associated to ψ̃ by the principle of functoriality. The formalism of [10] then yields

ψ : ^{L}T → ^{L}G

and if θ is the restriction of θ̃ to T (E) then ψ_{∗} takes {θ} to the set Π(ρ) and θ is associated to φ : W_{K/E} → ^{L}T.

Consider the set 𝔈(T) introduced in [12] and suppose κ is a homomorphism of 𝔈(T) into C^{×}, that is, a homomorphism of X_{∗}(T_{sc}) into C^{×} which is trivial on

X_{∗}(T_{sc}) ∩ (∑_{𝔊(K/E)}(σ − 1)X_{∗}(T)).

Then κ extends to a 𝔊(K/F)-invariant homomorphism κ′ of X_{∗}(T) into C^{×}. Since

LT^{0} = Hom(X_{∗}(T), C^{X})

the homomorphism κ′ is an element of ^{L}T^{0} which commutes with 𝔊(K/E) and hence with φ(W_{K/E}). Then ϵ = ψ(K′) commutes with ρ(W_{K/E}) and lies in S. Since ϵ is uniquely determined modulo this centre by κ, its image in S^{0}\S is uniquely determined.

If there is a non-trivial κ it is unique, and there exists a non-trivial character if and only if [𝔊(T)] = 2. However, it is easy to see that if T′ = T (E) ⊆ T̃(F) and if 𝔈(T′) is defined as in §2 then

𝔈(T) ≃ 𝔈(T′).

If κ is non-trivial then ϵ is represented by an element which is congruent to

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

modulo scalars.

For each L, T̃, θ̃ with [𝔈(T)] = 2 and

ρ̃ ∼ Ind (W_{K/F} , W_{K/L}, θ̃)

we obtain an ϵ, uniquely determined modulo the centre. When ρ̃ is fixed within its equivalence class then ϵ has the form (6.4) only after conjugation. If (a_{v}) lies in S then all the a_{v} have the same image in the projective group and lie in the projective centralizer of ρ̃(W_{K/F}). But the projective centralizer of ρ̃(W_{K/F}) consists of two or four elements, two when

θ̃(x̄)^{2} ≢ θ̃(x)^{2}

and four otherwise. If G is SL(2) so that [𝔈(T)] is always 2 it is easily seen that this centralizer consists of the identity and the ϵ obtained from non-isomorphic L, T̃, θ̃.

To show that in general S is formed of S^{0} and the S^{0}ϵ, where ϵ is obtained from the L, T̃, θ̃ with [𝔈(T)] = 2 we show that if we start from L, T̃, θ̃ with

ρ̃ = Ind(W_{K/F} , W_{K/L}, θ̃)

and form the corresponding ϵ̄ in the associate group of Res_{E/F} SL(2), which is

∏𝔊(K/F)\𝔊(K/E) P GL(2, C) × 𝔊(K/E),

and ϵ̄ lifts to ϵ in ^{L}G^{0} then ϵ lies in S only if [𝔈(T)] = 2. But ϵ is still the image of κ′ and if ϵ lies in S then κ′ is invariant under 𝔊(K/E) and its restriction to X_{∗}(T_{sc}) is non-trivial.

If ϵ lies in S^{0} we set ⟨ϵ, π⟩ = 1. Otherwise ϵ in S is associated to a T′ and a θ with [𝔈(T′)] = 2 and π ∈ Π(θ). We set ⟨ϵ, π⟩ equal to 1 or to −1 according as π ∈ Π^{+}(θ) or π ∈ Π^{−}(θ). We have already observed that ⟨ϵ, π⟩ is then multiplicative in ϵ. If to each π in Π(θ) we associate the character ϵ → ⟨ϵ, π⟩ we obtain a bijection from π(θ) to the dual of S^{0}\S.

Now we treat a global field F. We again suppose that A is A(E) where

A ⊆ Res_{F/E}GL(1).

If w is a place of E then over E_{w} the group on the right becomes

∏_{v|w} Res_{F_{v}/E_{w}}GL(1).

In order to remain within the framework of the earlier paragraph we suppose that over E_{w} the group A is also a product ∏ A_{v} with

A_{v} ⊆ Res_{F_{v}/E_{w}}GL(1),

but this is only a matter of convenience. The definition of S is now somewhat different. It is the set of all s such that for each place w there is an a_{w} in the centre of ^{L}G^{0} which is such that a_{w}s commutes with ρ(W_{E_{w}}). S^{0} can therefore be taken to be the connected component of S. The analysis of S^{0}\S may be repeated, because 𝔈(T′/A)/𝔈(T′/F)) may be identified by means of Tate-Nakayama duality with a quotient of X_{∗}(T_{sc}). If w is a place of E the local analogue of S is

S_{w} = ∏_{v|w} S_{v}

and the local analogue of S^{0} is

S_{w}^{0} = ∏_{v|w} S_{v}^{0}.

We clearly have a map ϵ → ϵ_{w} = Π_{v}ϵ_{v} of S^{0}\S to S_{w}^{0}\S_{w}. If

ρ : W_{K/E} → ^{L}G

then Π(ρ) consists of the representations π = ⊗ π_{v} where π_{v} ∈ Π(ρ_{v}) for all v and π_{v} contains the trivial representation of K′_{v} for almost all v. We introduce the pairing of S^{0}\S with Π(ρ) given by

⟨ϵ, π⟩ = Π_{v}⟨ϵ_{v}, π_{v}⟩.

The product is over all places of F and yields a surjection from Π(ρ) to the dual of S^{0}\S. The numbers m(π) appearing in Propositions 6.7 and 6.8 may be written as

\frac{d(π)}{[S : S^{0}]} ∑_{S^{0}\S}⟨ϵ, π⟩.

We have also to find another interpretation for d(π). π is π(ρ) where

ρ : W_{E} → ^{L}G.

Suppose we have another

σ : W_{E} → ^{L}G.

ρ is defined by ρ̃: W_{F} → ^{L}G̃ and, by Lemma 3 of [12], σ too is defined by σ̃ : W_{F} → ^{L}G̃. It is easy enough to deduce from the results of [14] that σ_{w} and ρ_{w} are conjugate under ^{L}G^{0} for all places w of E if and only if σ̃ ≃ ω ⊗ ρ̃ and σ̃ and ρ̃ are locally equivalent in the sense defined earlier. Thus local equivalence is more appropriately defined as the conjugacy of σ_{w} and ρ_{w} under ^{L}G^{0} for all w.

It also follows from [14] that ρ and σ themselves are conjugate under ^{L}G^{0} if and only if σ̃ ∼ ω ⊗ ρ̃ where ω is 1 on the group B of all x ∈ I_{F} such that for some finite extension K of F there is a y ∈ K^{×} with xy ∈ A(A_{K}). Since B contains F^{×}A(A_{E}) this is a stronger than global equivalence. ρ̃ and σ̃ are globally equivalent if and only if σ is conjugate under ^{L}G^{0} to

w → ω(w)ρ(w)

where w → ω(w) is a continuous locally trivial 1-cocycle of W_{E} with values in ^{L}A^{0}, the centre of LG^{0}. In any event we can define local and global equivalence and the integer d(π) entirely in terms of ρ and the associate group.

7. Quaternion algebras again. It is a straightforward matter to decompose the trace formula for a quaternion algebra into a stable and a labile part. Let G̃ be the group defined by a quaternion algebra over the global field F, let A be as in §5 with A ⊆ Nm G̃(A), and let

G′ = {g ∈ G̃(A)|Nm g ∈ A}.

Otherwise our notation will be along the lines of §5. We want to evaluate the trace of the representation s on the space of functions φ which satisfy

φ(zg) = χ^{−1}(z)φ(g) z ∈ _{O}Z′

and which are square integrable on _{0}Z′G′_{F} \G′. If f is a function on G(A) with

f(g) = Π_{v}f_{v}(g_{v}),

where the f_{v} satisfy the usual conditions, in particular

f_{v}(zg_{v}) = χ(z)f_{v}(g_{v}), z ∈ _{0}Z′_{v},

then

s(f)φ(h) = ∫_{_{0}Z′\G′} φ(hg)f(g)dg.

The results and definitions of §4 clearly have analogues for the functions f_{v}, and we shall employ them.

The first and trivial term of the trace formula is

∑_{γ∈_{0}Z_{F′} \Z_{F′}} f(γ).

The second term will be broken up immediately into a stable and a labile part. If μ(T′) is the order of the kernel of

𝔈(T′/F) → 𝔈(T′/A)

then the stable part is

\frac{1}{2} ∑_{T′} \frac{μ(T′)}{[𝔈(T′/A) : Im 𝔈(T′/F)]} meas (_{0}Z′T′_{F} \T′) ∑_{γ}^{′} Φ^{T′/1}(γ, f)

with

Φ^{T′/1}(γ, f) = Π_{v}Φ^{T′/1}(γ, f_{v}).

The outer sum is over a set of representatives for the stable conjugacy classes of Cartan subgroups. The inner sum is over the regular elements of T′_{F} modulo _{0}Z′_{F}. The labile part is a sum over representatives of those stable conjugacy classes for which

[𝔈(T′/A) : Im 𝔈(T′/F)] = 2

of

(7.1)  \frac{1}{4} ∑_{γ} μ(T′) meas (_{0}Z′T′_{F} \T′)Φ^{T′}(γ, f).

We note also that

𝔇(T′/F) = NmD^{×}/A_{F} NmL^{×}

and that it is implicit in the above discussion that NmD^{×} consists of the elements of F^{×} that are positive at every real place where D does not split. The sum in (7.1) is at first only over the regular elements in T′_{F} modulo _{0}Z′_{F} but we may extend it over all of T′_{F} for the additional terms will all be 0.

Let H̃ be GL(2) over F and define H′ accordingly. For almost all v,

H′_{v} ≃ G′_{v}

and we may choose the isomorphism so that the maximal compact subgroups correspond. For these v let ϕ_{v} be the image of f_{v} by the isomorphism. For the v at which H̃ does not split define ϕ_{v} as in §4. An easy comparison, as in §16 of [6], shows that the stable parts of trace s(f) and trace r(ϕ) are equal. We write the first as

∑_{π} n(π) trace π(f)

and the second as

∑_{τ} n(τ) trace τ(ϕ).

The argument used to prove Lemma 6.1 and the remarks of §4 imply that if τ lies in the L-indistinguishable class corresponding to that of π then

n(π) = n(τ)Π_{v}c(π_{v}).

Here c(π_{v}) is 1 when H̃_{v} is split.

We first study the labile part for a particular G′. If F′ is a given non-archimedean local field and L′ a given quadratic extension of F′ we choose a totally real field F and a totally imaginary quadratic extension of it so that for some place v of F the pair F_{v}, L_{v} is isomorphic to F′, L′. Choose a quaternion algebra which splits at every finite place except v, and let

A = Nm I_{L}.

There is only one stable conjugacy class of Cartan subgroups with

[𝔈(T′/A) : Im 𝔈(T′/F)] = 2

and the labile part of the trace formula is the sum over the characters of T′_{F} \T′ which equal χ^{−1} on _{0}Z′ of

(7.2)  \frac{1}{4} Θ_{v}(f_{v})Π_{w≠v}(∑_{π_{w}∈Π^{+}(θ_{w})} χ_{π_{w}}(f_{w}) − ∑_{π_{w}∈Π^{−}(θ_{w})} χ_{π_{w}}(f_{w})).

We have fixed a non-trivial character ψ of F \A and at a place where the quaternion algebra splits the sets Π^{+}(θ_{w}) and Π^{−}(θ_{w}) are defined with respect to ψ_{w}. Θ_{v} is defined with respect to ψ_{v} as in §4. At an archimedean place w where the algebra does not split Π(θ_{w}) consists of a single element π. We place it in Π^{+}(θ_{w}) or Π^{−}(θ_{w}) according as Θ_{w} equals +χ_{π} or −χ_{π}. One of the two sets remains empty.

Define θ̄ by

θ̄(γ̄) = θ(γ)

where γ → γ̄ is the involution. If θ̄_{v} = θ_{v} then Θ_{v} = 0. Otherwise replacing θ by θ̄ changes the sign of an even number of factors in (7.2) but does not change the expression itself. We sum over pairs \{θ, θ̄\} with θ ≠ θ̄ and replace the \frac{1}{4} by \frac{1}{2}.

It is clear that Θ_{v} is a finite linear combination of irreducible characters

Θ_{v} = ∑ a_{i}χ_{π_{v}^{i}}.

Consider

π = π_{v}^{i} ⊗ (⊗_{w≠v}π_{w})

with π_{w} in Π(θ_{w}). If m(π) is the multiplicity with which π occurs in s then

m(π) = n(π) ± a_{i}/2.

Varying the π_{w} within Π(θ_{w}) does not change n(π) but it does change the sign. We conclude that a_{i} is an integer. Moreover if a_{i} ≠ 0 we may arrange that m(π) be positive. Since π must lie in the restriction to G′ of an automorphic representation of G̃(A) we conclude from the strong form of the multiplicity one theorem that π_{v}^{i} ∈ Π(θ_{v}) when a_{i} ≠ 0. We know that Π(θ_{v}) consists of two elements when θ̄_{v} ≠ θ_{v}.

The orthogonality relations for characters of G′_{v} show that

\frac{1}{2} ∑_{T′_{v}} \frac{[𝔇(T′_{v})]}{meas Z′_{v}\T′_{v}} ∫_{Z′_{v}\T′_{v}} |Θ_{v}(γ)|^{2}Δ^{2}(γ)dγ = ∑ a_{i}^{2}.

The sum is over a set of representatives for stable conjugacy classes of Cartan subgroups of G′_{v}. However, the formula for Θ_{v} shows immediately that the left side is 2 when θ̄_{v} ≠ θ_{v}. Thus a_{i} = ±1. Since there are two a_{i} and Θ_{v} is not stable, they must have opposite signs, and with an appropriate choice of Π^{+}(θ_{v}) and Π^{−}(θ_{v}) we have

(7.3)  Θ_{v} = ∑_{π_{v}∈Π^{+}(θ_{v})} χ_{π_{v}} − ∑_{π_{v}∈Π^{−}(θ_{v})} χ_{π_{v}}.

More generally if Ḡ′_{v} is defined by Ā_{v} and if T̄′_{v} belongs to the stable conjugacy class associated to L_{v} then

[𝔇(T̄′_{v})] = 2

if and only if Ā_{v} ⊆ NmL_{v}^{×} and Ḡ′_{v} ⊆ G′_{v}. If π_{v}^{+} and π_{v}^{−} are the restrictions of the elements of Π^{+}(θ_{v}) and Π^{−}(θ_{v}) then Θ_{v} is equal to

χ_{π_{v}^{+}} − χ_{π_{v}^{−}}

on Ḡ′_{v}. If θ̄_{v} and θ_{v} are not equal on T̄′_{v} then π_{v}^{+} and π_{v}^{−} are not equivalent. Otherwise they are and c(π_{v}^{+}) ≧ 2. We know already that c(π_{v}^{+}) ≦ 2.

We can summarize the local results.

Lemma 7.1. Suppose F is local field. The L-indistinguishable class of π consists of 1 or 2 elements. It consists of two elements if and only if π lies in Π(θ), where θ is a character of T′, [𝔇(T′)] = 2, and θ(γ) ≢ θ(γ̄). Moreover c(π) is 1 unless π lies in Π(θ), where [𝔇(T′)] = 2, and θ(γ) ≡ θ(γ̄) when c(π) = 2.

When Π(θ) consists of two elements we may decompose it into Π^{+}(θ) and Π^{−}(θ) in such a way that

Θ = ∑_{π∈Π^{+}(θ)} χ_{π} − ∑_{π∈Π^{−}(θ)} χ_{π}.

If θ is replaced by θ̄ then Θ is replaced by −Θ, and consequently Π^{+}(θ̄) = Π^{−}(θ).

If F is a global field and θ a character of T′_{F} \T′ we let Π(θ) be the set of tensor products ⊗ π_{v} with π_{v} in Π(θ_{v}) for all v. It is understood that in such a tensor product π_{v} contains the trivial representation of the maximal compact for almost all v.

Proposition 7.2. Suppose π = ⊗ π_{v} is contained in no Π(θ) where θ is a character of T′_{F} \T′ and

[𝔇(T′/A) : Im 𝔇(T′/F)] = 2.

Let τ be in the L-indistinguishable class of H′ corresponding to that of π. For all g ∈ Ḡ(A)

m(π^{g}) = m(π)

and

m(π) = n(τ)Π_{v}c(π_{v}).

Now suppose π belongs to Π(θ) and [𝔇(T′/A) : Im 𝔇(T′/F)] = 2. The type of π is again defined to be that of θ, either (a) or (b). Let τ be a representation of H′ whose L-indistinguishable class is that of π. It is reasonable to set

d(π) = d(τ).

Suppose π is of type (a). We introduce the group consisting of two elements 1, ϵ. At a place v where the quaternion algebra splits we define ⟨1, π_{v}⟩ to be 1 and ⟨ϵ, π_{v}⟩ to be as before. At a place v where the algebra does not split but θ_{v} ≠ θ̄_{v} set ⟨1, π_{v}⟩ = 1 and ⟨ϵ, π_{v}⟩ equal to +1 or −1 according as π_{v} ∈ Π^{+}(θ_{v}) or π_{v} ∈ Π^{−}(θ_{v}). Observe that ⟨ϵ, π_{v}⟩ depends on θ_{v}. At a place where the quaternion algebra does not split and θ_{v} = θ̄_{v} we set ⟨1, π_{v}⟩ = 2 and ⟨ϵ, π_{v}⟩ = 0. Let

⟨1, π⟩ = Π_{v}⟨1, π_{v}⟩

and

⟨ϵ, π⟩ = Π_{v}⟨ϵ, π_{v}⟩.

Proposition 7.3. If π is of type (a) then

m(π) = \frac{d(π)}{2}(⟨1, π⟩ + ⟨ϵ, π⟩).

If θ_{v} = θ̄_{v} for some v then

m(π) = n(π) = n(τ) ∏_{v} c(π_{v}) = \frac{d(π)}{2} ∏_{v} c(π_{v}) = \frac{d(π)}{2}⟨1, π⟩

and ⟨ϵ, π⟩ = 0. If θ_{v} ≠ θ̄_{v} for all v then

m(π) = n(π) + \frac{d(π)}{2}⟨ϵ, π⟩ = \frac{d(π)}{2}(⟨1, π⟩ + ⟨ϵ, π⟩).

If π is of type (b) we introduce a group consisting of four elements 1, ϵ_{1}, ϵ_{2}, ϵ_{3} with ϵ_{i}^{2} = 1. The numbers ⟨ϵ_{i}, π_{v}⟩ are defined just as those ⟨ϵ, π_{v}⟩ of the previous lemma were and

⟨ϵ_{i}, π⟩ = Π_{v}⟨ϵ_{i}, π_{v}⟩.

Since θ^{i} = θ̄^{i}, i = 1, 2, 3 and π ∈ Π(θ), the algebra is split at the infinite places and not split at some finite places. Consequently

⟨ϵ_{i}, π⟩ = 0 i = 1, 2, 3.

Proposition 7.4. If π is of type (b) then

m(π) = \frac{d(π)}{4}(⟨1, π⟩ + ⟨ϵ_{1}, π⟩ + ⟨ϵ_{2}, π⟩ + ⟨ϵ_{3}, π⟩).

For such a π,

m(π) = n(π) = n(τ) ∏_{v} c(π_{v}) = \frac{d(π)}{4}⟨1, π⟩.

When G′ is defined by a connected subgroup of Res_{F/E}GL(1), we may interpret the groups appearing in these two propositions as S^{0}\S, just as in the previous paragraph.

8. Afterword. There is a condition implicit when we take the group G′ to be G(A_{E}) where G is a subgroup of Res_{F/E}G̃ defined as the inverse image of the subgroup A of G_{1} = Res_{F/E}GL(1). A is of course taken to be connected and

A(A_{E}) = ∏_{w} A(E_{w}),

the product being restricted, and

A(E_{w}) ⊆ G_{1}(E_{w}) = ∏_{v|w} F_{v}^{×}.

However, A(E_{w}) need not be a product

∏_{v|w} A_{v}

with A_{v} ⊆ F_{v}^{×}, and we may not be free to apply the results of the early paragraphs.

We might have developed the local theory of L-indistinguishability for the groups G(E_{w}). However 𝔈(T/E_{w}) may no longer be of order 2. All of its characters would have to be considered, and some of them give rise to groups H which are not abelian. The local theory would provide a linear transformation from stable distributions on H to distributions on G. Although not difficult it would have been elaborate, and unnecessary for the global theory.

In the global theory it is the quotient

𝔈(T/A_{E})/Im 𝔈(T/E)

which is central. If T is a Cartan subgroup of G associated to the quadratic extension L it is again

I_{F} /A(A_{E})F^{×} Nm I_{L}

and of order 1 or 2. The local theory need only be developed for the character of 𝔈(T/E_{w}) obtained by pulling back the non-trivial character of the quotient via

𝔈(T/E_{w}) → 𝔈(T/A_{E}) → 𝔈(T/A_{E})/Im 𝔈(T/E).

The necessary results are easily deduced from §2 - §4.

The principal results of §5 - §7 remain valid, and the proofs are the same. We remark only that S_{w} is no longer a product Π_{v|w}S_{v}.

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Faculté des Sciences, Dijon, France; Institute for Advanced Study, Princeton, New Jersey