Some results on the admissible representations of non-connected reductive p-adic groups

Ann. scient. kc. Nom. Sup., 4e sCrie, t. 30, 1997, p. 97 ?I 146.

SOME RESULTS ON THE ADMISSIBLE REPRESENTATIONS OF

NON-CONNECTED REDUCTIVE p-ADIC GROUPS

BY DAVID GOLDBERG (*) AND REBECCA HERB (**)

ABSTRACT. - We examine induced representations for non-connected reductive p-adic groups with G/Go abelian. We describe the structure of the representations Ind$, (u), P” a parabolic subgroup of Go and c a discrete series representation of the Levi component of PO. We develop a theory of R-groups, extending the theory in the connected case. We then prove some general representation theoretic results for non-connected p-adic groups with abelian component group. The notion of cuspidal parabolic for G is defined, giving a context for this discussion, Intertwining operators for the non-connected case are examined and the notions of supercuspidal and discrete series are defined. Finally, we examine parabolic induction from cuspidal parabolic subgroups of G. We develop a theory of R-groups, and show these groups parameterize the induced representations in a manner consistent with the connected case and with the first set of results as well.

1. Introduction

The theory of induced representations plays a fundamental role within representation theory in general. Within the theory of admissible representations of connected reductive algebraic groups over local fields, parabolic induction is used to complete classification theories, once certain families of representations are understood [3], [8], [9], [lo]. The theory of admissible representations on non-connected reductive groups over nonarchimedean local fields has been addressed in part in [l], [4], [6], [ll], among other places. We will study certain aspects of parabolic induction for disconnected groups whose component group is abelian.

Let F be a locally compact, non-discrete, nonarchimedean field of characteristic zero. Let G be a (not necessarily connected) reductive F-group. Thus G is the set of F-rational points of a reductive algebraic group defined over F. Let Go be the connected component of the identity in G. We assume that G/Go is finite and abelian.

Our goal is to address three major points. The first is an extension of the results of [6] to the case at hand. This entails a study of induction from a parabolic subgroup of G”

(*) Partially supported by National Science Foundation Fellowship DMS9206246 and National Science Foundation Career Grant DMS9501868. Research at MSRI supported in part by National Science Foundation Grant DMS9022140.

(**) Partially supported by National Science Foundation Grant DMS9400797.

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98 D.GOLDBERG AND R. HERB

to G. In particular, suppose that P” = AP’N is a parabolic subgroup of G”, and let co be an irreducible discrete series representation of M”. We are interested in the structure of ~0 = Ind& (00). In [l] Arthur suggests a construction, in terms of the conjectural local Langlands parameterization, of a finite group whose representation theory should describe the structure of ~0, when G/G” is cyclic. In [6] the case where G/G” is of prime order is studied, and there is a construction, on the group side, of a finite group &(~a) which (along with an appropriate 2-cocycle) parameterizes the components of ~0. It is also shown there that RG(cO) must be isomorphic to Arthur’s group Rvi,go, if the latter exists. One cannot confirm the existence of Rti,,,, without proofs of both the local Langlands conjecture and Shelstad’s conjecture [ 121 that R,, ,00 is isomorphic to RG~ (oo) . (See [l] for the precise definitions of Rw,,, and R,,,,,, ) Here, by extending the definition of the standard intertwining operators (c$ Section 4) we show we can construct a group RG(oO) in a manner analogous to [6], and show that it has the correct parameterization properties (cc Theorems 4.16 and 4.17). An argument, similar to the one given in [6] shows that if G/G” is cyclic, then Ra(oO) must be isomorphic to R+,,,; assuming the latter exists (~5 Remark 4.18).

The second collection of results is an extension of some standard results in admissible representation theory to the disconnected group G. In order to develop a theory consistent with the theory for connected groups, one needs to determine an appropriate definition of parabolic subgroup. There are several definitions in the literature already, yet they do not always agree. We use a definition of parabolic subgroup which is well suited to our purposes. Among the parabolic subgroups of G we single out a collection of parabolic subgroups which we call cuspidal. They have the property that they support discrete series and supercuspidal representations and can be described as follows. Let P” be a parabolic subgroup of Go with Levi decomposition M”N and let A be the split component of MO. Let &l = CG(A). Then P = MN is a cuspidal parabolic subgroup of G lying over PO. We also say in this case that M is a cuspidal Levi subgroup of G. Thus cuspidal parabolic subgroups of G are in one to one correspondence with parabolic subgroups of Go.

Using our definitions we can prove the following. Let M be a Levi subgroup of G and let M” = M n Go.

LEMMA 1.1. - (i) If M is not cuspidal, then M has no supercuspidal representations, i.e., admissible representations with matrix coefficients which are compactly supported mod&o the center of M and have zero constant term along the nil radical of any proper parabolic subgroup of M.

(ii) If M is cuspidal and T is an irreducible admissible representation of M, then IT is supercuspidal if and only if the restriction of T to M” is supercuspidal.

(iii) If M is not cuspidal, then M has no discrete series representations, i.e. unitary representations with matrix coejlkients which are square-integrable modulo the center of M.

(iv) If M is cuspidal and T is an irreducible unitary representation of M, then T is discrete series if and only if the restriction of T to MO is discrete series.

Using Lemma 1.1 it is easy to extend the following theorem from the connected case to our class of disconnected groups.

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SOME RESULTS ON THE ADMISSIBLE REPRESENTATIONS 99

THEOREM 1.2. - Let r be an irreducible admissible (respectively tempered) representation of G. Then there are a cuspidal parabolic subgroup P = MN of G and an irreducible supercuspidal (respectively discrete series) representation o of M such that T is a subrepresentation of Ind): (c).

Let PI = MI Ni and P2 = MzN, be cuspidal parabolic subgroups and let ffi be irreducible representations of Mi, i = 1,2, which are either both supercuspidal or both discrete series. By studying the orbits for the action of PI x PZ on G, we are able to extend the proof for the connected case to our situation and obtain the following theorem.

THEOREM 1.3. - Let PI = MI N1, P2 = Mz N2, ol, 02 be as above. Then if~r = Indgl (rl) and 7~2 = Indg2(a2) have a nontrivial intertwining, then there is y E G so that

M2 = yMIy-1 and u2 N yu2y-‘.

The third question of study is the structure of x = Ind$(a) when P = MN is a cuspidal parabolic subgroup of G and 0 is a discrete series representation of M. We show that, as in the connected case, the components of n are naturally parameterized using a finite group R. As in the connected case we first describe a collection of standard intertwining operators R( w, (T) which are naturally indexed by w E WC (a) = NG (a)/M, where

NG(u) = {x E N,(M) : ox N o}.

We prove that there is a cocycle q so that

Let go be an irreducible component of the restriction of (T to MO, and P” = M”N = P n Go. Then g c IndEo (go) so that

r = Indg(g) C Ind~(Ind$(ao)) N Ind$,(ao).

Using the intertwining operators and R-group theory developed earlier for the representation Indza ( ao), we can prove the following results. First, the collection {R(w) o) , w E WC (a)} spans the commuting algebra of rr. Second, let Qt be the set of positive restricted roots for which the rank one Plancherel measures of ~0 are zero and let W((ar) be the group generated by the reflections corresponding to the roots in Cpi. Then W(@r) is naturally embedded as a normal subgroup of wG((T) and WG(g) is the semidirect product of IV($) and the group

R, = {w E wG(o) : WQ > 0 for all ci E (a:}.

Finally, R(w, a) is scalar if w E IV(@). This proves that the operators R(w, a), w E R,, span the intertwining algebra. But we can compute the dimension of the space of intertwining operators for Indg (a), again by comparison with that of IndFo (go), and we find that it is equal to [R,]. Thus we have the following theorem.

THEOREM 1.4. - The R( w, a), w E R,, form a basis for the algebra of intertwining operators of Indg (c).


Just as in the connected case, we show that there are a finite central extension

over which n splits and a character x,, of 2, so that the irreducible constituents of Indg (a) are naturally parameterized by the irreducible representations of R, with Z,-central character x~.

Finally, we give a few examples which point out some of the subtleties involved in working with disconnected groups. For instance, we show that if we do not restrict ourselves to cuspidal parabolic subgroups, then the standard disjointness theorem for induced representations fails. Examples such as these show why one must restrict to induction from cuspidal parabolic subgroups in order to develop a theory which is consistent with that for connected groups.

Many interesting problems involving disconnected groups remain. For example, the question of a Langlands classification is still unresolved, and some of the results here on intertwining operators may help in this direction, One also hopes to remove the condition that G/G” is abelian, and extend all the results herein to that case. Problems such as these we leave to further consideration.

The organization of the paper is as follows. In $2 we give the definition of parabolic subgroup and prove Lemma 1.1 and Theorem 1.2. The proof of Theorem 1.3 is in §3. The results on induction from a parabolic subgroup of Go to G are in $4, and the results on induction from a parabolic subgroup of G to G, including Theorem 1.4, are in 55. Finally, $6 contains examples that show what can go wrong when we induce from parabolic subgroups of G which are not cuspidal.

The first named author would like to thank the Mathematical Sciences Research Institute in Berkeley, California, for the pleasant and rich atmosphere in which some of the work herein was completed.

2. Basic definitions

Let F be a locally compact, non-discrete, nonarchimedean field of characteristic zero. Let G be a (not necessarily connected) reductive F-group. Thus G is the set of F-rational points of a reductive algebraic group over F. Let Go be the connected component of the identity in G. We assume that G/Go is finite and abelian.

The split component of G is defined to be the maximal F-split torus lying in the center of G. Let A be any F-split torus in G and let iV = CG(A). Then Ii4 is a reductive F-group. Now A is called a special torus of G if A is the split component of 1M. (Of course A is an F-split torus lying in the center of A&. The only question is whether or not A is maximal with respect to this property.)

LEMMA 2.1. - Let A be a special torus of Go. Then A is a special torus of G.

Proof. - Let 1M = Cc(A) and M” = CQ(A) = A! II Go. Write Z(1M) and Z(M”) for the centers of A4 and MO respectively. Now A is the maximal F-split torus lying in Z(M”) and A c Z(M). Suppose A’ is the maximal F-split torus lying in Z(M). Then

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SOME RESULTS ONTHE ADMISSIBLE REPRESENTATIONS 101

A c A’. But A’ is a torus, so it is connected. Hence A’ c Z(M) tl MO c Z(M”). Thus A’ c A and so A’ = A is the split component of M. n

REMARK 2.2. - The converse of Lemma 2.1 is not true. For example, let G = O(2) = SO(2) u t&O(2) h w ere SO(2) N FX is the group of 2 x 2 matrices

,aEFX, andw=

satisfies wd(a)w-’ = d(a-‘),a E FX. Let A = {d(l)}. Then M = CG(A) = G and Z(M) = {M(l)}. Thus A is a special torus of G. However Mo = C,y(A) = Go and Z(MO) = Go is an F-split torus. Hence A is not the maximal F-split torus in Z(M”) and so is not special in Go.

If G is connected, then A is a special torus of G by the above definition if and only if A is the split component of a Levi component M of a parabolic subgroup of G. We will define parabolic subgroups in the non-connected case so that we have this property in the non-connected case also.

Let A be a special torus of G and let A4 = CG(A). Then M is called a Levi subgroup of G. The Lie algebra L(G) can be decomposed into root spaces with respect to the roots + of L(A):

-V? = W)o @ c L(G)a QECJ

where L(G)0 is the Lie algebra of M. Let @+ be a choice of positive roots, and let N be the connected subgroup of G corresponding to CaE@+ L(G),. . Since elements of M centralize A and L(A), they preserve the root spaces with respect to L(A). Thus M normalizes N. Now P = MN is called a parabolic subgroup of G and (P, A) is called a p-pair of G. The following lemma is an immediate consequence of this definition and Lemma 2.1.

LEMMA 2.3. - Let PO = M”N be a parabolic subgroup of Go and let A be the split component of MO. Let M = Ca(A). Then P = MN is a parabolic subgroup of G and P n Go = PO.

LEMMA 2.4. - Let P be a parabolic subgroup of G. Then PO = P n Go is a parabolic subgroup of Go.

Proof. - Let P = MN be a parabolic subgroup of G and let A be the split component of M. Let nil0 = CGO(A) = MnGO. Let Al be the split component of MO. Then A c Al so that CGO(A~) C CGO(A) = MO. But Al is in the center of MO, so that M” c CGo(A1). Thus CGO(A~) = MO so that Al is a special torus in Go and MO is a Levi subgroup of Go. Let Q, and @r denote the sets of roots of L(A) and L(A,) respectively. For each or E +I, the restriction ror of al to L(A) is non-zero since CGO (A,) = CGo(A) = MO. Let @+ be the set of positive roots used to define N. Then (at = {or E a1 : rol E (a+} is a set of positive roots for @r and

Thus P” = M”N is a parabolic subgroup of Go. n

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102 D.GOLDBERG AND R.HERB

We say the parabolic subgroup P of G lies over the parabolic subgroup P” of Go if P” = P n Go. We will also say the Levi subgroup M of G lies over the Levi subgroup M” of Go if M” = A4 n Go. Lemma 2.4 and its proof show that every parabolic (resp. Levi) subgroup of G lies over a parabolic (resp. Levi) subgroup of Go.

REMARK 2.5. - There can be more than one parabolic subgroup P of G lying over a parabolic subgroup P” of Go. For example, define G = O(2) and Go = SO( 2) as in Remark 2.2. Then A = (d(1)) and A0 = SO(2) are special vector subgroups of G corresponding to parabolic subgroups P = O(2) and PO = SO(a) respectively. Both lie over the unique parabolic subgroup SO(2) of Go.

LEMMA 2.6. - Let P” = M”N be a parabolic subgroup of Go and let A be the split component of MO. Let M = CG(A) and let P = MN. Then if PI is any parabolic subgroup of G lying over PO we have P c PI. Further, M is the unique Levi subgroup lying over MO such that the split component of M is equal to A.

Proof. - Write PI = MI N where MI lies over M ‘. Let Al be the split component of Ml. Then A1 c A so that M = Cc(A) c CG(AI) = Ml. Clearly Ml = M if and only if AI = A. n

REMARK 2.7. - Lemma 2.6 shows that there is a unique smallest parabolic subgroup P of G lying over P ‘. Although it is defined using a Levi decomposition P” = M”N of PO, it is independent of the Levi decomposition. Recall that if MF and M: are two Levi components of P* with split components Al and A2 respectively, then there is n E N such that AZ = nAln-’ and Mi = nMfn-‘. NOW if Mi = GG(Ai),i = 1,2, we have M2 = CG(A~) = nCG(Al)n-l = nMIn-‘, and MzN = nMInelN = MIN since MI normalizes N.

Let 2 be the split component of G. We let Cp (G, 2) denote the space of all smooth complex-valued functions on G which are compactly supported modulo 2. We say f E C,-(G, 2) is a cusp form if for every proper parabolic subgroup P = MN of G,

J’ N f(zn)dn = ” Vz E G.

Let 'd(G) denote the set of cusp forms on G. We say G is cuspidal if 'd(G) # (0). We know that every connected G is cuspidal.

LEMMA 2.8. - G is cuspidal if and only if the split component of G is equal to the split component of Go. Moreover, if G is cuspidal, then a subgroup N of G is the nilradical of a proper parabolic subgroup of G if and only if N is the nilradical of a proper parabolic subgroup of Go. If G is not cuspidal, then G has a proper parabolic subgroup G1 with nilradical Nl = (1).

Proof. - First suppose that G and Go have the same split component 2. Let f # 0 E ‘d(G’). Define F : G 3 C by F(x) = f(z),x E Go, and F(z) = 0,~ $Z Go. Then F E C,-(G, 2) and is non-zero. Let P = MN be any proper parabolic subgroup

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of G. Then N c Go, so for all n E N,x E G, xn E Go if and only if x E Go. Thus for x 6 Go,

s F(xn)dn = 0

N

while for x E Go,

s F(xn)dn =

N , I N f(xnW.

Now P” = P n Go = MON is a parabolic subgroup of Go. Suppose that PO = Go. Then P lies over Go so that by Lemma 2.6, G c P. This contradicts the fact that P is a proper parabolic subgroup of G. Thus P” = M”N is a proper parabolic subgroup of Go. Since f is a cusp form for Go we have

s N f(xn)dn = ” bfx E Go.

Thus F is a non-zero cusp form for G. The above argument also showed that if P = MN is a proper parabolic subgroup of

G, then P” = M”N is a proper parabolic subgroup of Go. Conversely, if P” = M”N is a proper parabolic subgroup of Go and P = MN is any parabolic subgroup of G lying over PO, then clearly P # G.

Conversely, suppose that G and Go do not have the same split component. Let 2 be the split component of Go and define Gr = CG(Z). By Lemma 2.6, Gr is a proper parabolic subgroup of G. Further since Gr lies over Go its nilradical is N1 = { 1). Now if F is any cusp form on G and x E G, we have

F(x) = J

F(xn)dn = 0. Nl

Thus G has no non-zero cusp forms and so is not cuspidal. n

EXAMPLE 2.9. - Let G = O(2) as in Remarks 2.2 and 2.5. Then SO(2) is a cuspidal parabolic subgroup of G and O(2) is not cuspidal.

We can sum up the proceeding lemmas in the following proposition.

PROPOSITION 2.10. - Let PO = M”N be a parabolic subgroup of Go. Then there is a unique cuspidal parabolic subgroup P = MN of G lying over PO. It is contained in every parabolic subgroup of G lying over PO, and is dejined by M = Cc (A) where A is the split component of MO.

Now that we have parabolic subgroups of G, we want to study parabolic induction of representations. Many of the most basic notions of representation theory are defined in [13], chapter 1 for any totally disconnected group. In particular, admissible representations of G are defined and the following is an easy consequence of the definition.

LEMMA 2.11. - Let lI be a representation of G. Then II is admissible if and only if II IGo, the restriction of II to Go, is admissible.

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104 D. GOLDBERG AND R. HERB

Further, the results of Gelbart and Knapp regarding induction and restriction between a totally disconnected group G and an open normal subgroup H with G/H finite abelian can be applied to G and G ‘. If r is any admissible representation of Go on V, we will let Ind$ (7r) denote the representation of G by left translations on %={f:G-+V: f&o) = +&lf(g),b E G,go E Go).

LEMMA 2.12 (Gelbart-Knapp [5]). - Let II be an irreducible admissible representation of G. Then II]Q is a finite direct sum of irreducible admissible representations of Go. Let n-

be an irreducible constituent of II] GO which occurs with multiplicity r. Then

l&o cv r c TTY sG/Gr

where G, = {g E G : KY N 7r).

LEMMA 2.13 (Gelbart-Knapp [5]). - Let x be an irreducible admissible representation of Go. Then there is an irreducible admissible representation II of G such that x occurs in the restriction of II to Go with multiplicity r > 0. Let X denote the group of unitary characters of G/Go and Zet X(n) = {x E X : II @ x N II}. Then

is the decomposition of Ind$ (r) into irreducibles and r”[X/X(II)] = [G,/GO]. The following result was proved by Gelbart and Knapp in the case where the restriction

is multiplicity one [5]. Tadic [ 141 refined their result in the connected case. We now prove the more general result.

LEMMA 2.14. - Suppose that G is a totally disconnected group, and H is a closed normal subgroup, with G/H a finite abelian group. If II, and II2 are irreducible admissible representations of G, which have a common constituent upon restriction to H, then II2 N II, 18 X, for some character x with x]u F 1.

Proof - If the multiplicity of the restrictions is one, then this result holds by Gelbart- Knapp [5]. In particular, if [G/HI ’ p is rime, the statement is true. We proceed by induction. We know the Lemma holds when IG/HI = 2. Suppose the statement is true whenever IG1/HI < n. Suppose /G/HI = n. We may assume n is composite, so write n = km, with 1 < k < n. Let H c Gi c G with (G/Gil = k. If II~]G~ and II~]G, have a common constituent, then, by our inductive hypothesis, there is a x with x]cl E 1 with II2 N II1 @ x. Since (G/Gij c (G/Hj, we are done, in this case.

Now suppose that r is an irreducible subrepresentasion of both II1 ]H and II2 1~. Then, there are constituents Ri c II; IG1 so that Q- c Ri 1~. By the inductive hypothesis R2 = Ri @x, for some x of G1 whose restriction to H is trivial. But, since Gl/H c G/H is abelian we can extend x to a character 71 of G/H. Note that (IIi@v)(Gi has flr@x II 02 as a constituent, so, as we have seen above, II1 @ q @ w II IIs, for some character w of G whose restriction to G1 is trivial. Thus, the statement holds by induction. n

Let (x, V) be an admissible representation of G and let A(r) denote its space of matrix coefficients. We say rr is supercuspidal if A(K) c ‘A(G). Of course if G is not cuspidal,

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then Od(G) = (0) so that G has no supercuspidal representations. If P = MN is any parabolic subgroup of G, define V(P) = V(N) to be the subspace of V spanned by vectors of the form ~(n)w - u, w E V,n E N. Then we say rr is J-supercuspidal if V(P) = V for every proper parabolic subgroup P of G. If G is not cuspidal, then by Lemma 2.8 there is a proper parabolic subgroup Gi of G with nilradical Ni = { 1). For any admissible representation (x, V) of G, V(G,) = V(N,) = (0) # V so that x is not J-supercuspidal. Thus G has no J-supercuspidal representation.

Suppose now that G is cuspidal and let (x, V) be an irreducible admissible representation of G. Let (rro, V) denote the restriction of rr to Go.

LEMMA 2.15. - Assume that G is cuspidal. Then x is J-supercuspidal if and only if ~0 is J-supercuspidal if and only if any irreducible constituent of ~0 is J-supercuspidal.

Proof. - Since G is cuspidal, by Lemma 2.8 the set of nilradicals of proper parabolic subgroups is the same for both G and Go. Thus x is J-supercuspidal if and only if rro is J-supercuspidal. Moreover, since by Lemma 2.12 the irreducible constituents of r. are all conjugate via elements of G, it is clear that ~0 is J-supercuspidal if and only if every irreducible constituent of rri~g is J-supercuspidal if and only if any irreducible constituent of ~0 is J-supercuspidal. n

LEMMA 2.16. - Assume that G is cuspidal. Then K is supercuspidal if and only if 7ro is supercuspidal if and only if any irreducible constituent of ~0 is supercuspidal.

Proof. - Let 2 denote the split component of G. By Lemma 2.8 it is also the split component of Go.

Assume that rr is supercuspidal. Thus d(r) c ‘d(G). Let fo be a matrix coefficient of ~0. Then there is a matrix coefficient f of z so that f. is the restriction of f to Go. Now f E ‘d(G). S’ mce f smooth and compactly supported modulo 2, so is fo. Further, by Lemma 2.8 the nilradicals of proper parabolic subgroups are the same for both G and Go. Thus fa will satisfy the integral condition necessary to be a cusp form on Go. Hence d(no) c ‘d(G’) so that 7ro is supercuspidal.

Conversely, suppose that rra is supercuspidal. Let ~1 be an irreducible constituent of TO. Then r c Ind’&(ri) so that every matrix coefficient of rr is a matrix coefficient of the induced representation. But since Go is a normal subgroup of finite index in G, the restriction of Ind&(ri) to Go is equivalent to &G,GO rr:. Thus matrix coefficients of the induced representation can be described as follows. Let f be a matrix coefficient of Ind$(7ri) and fix g E G. Then there are matrix coefficients fz of rr:, z E G/Go, (depending on both f and g) so that for all go E Go,

f(m) = c fz(g0). XEG/GO

Since rl is supercuspidal, so is rr: for any 2 E G/Go, and so each fz E Od(G’). Thus the restriction of f to each connected component of G is smooth and compactly supported modulo 2. Also, if N is the nilradical of any proper parabolic subgroup of G, then

S, f(gn)dn = C / fz(n)dn = 0 xEG/GO N

since by Lemma 2.8, N is also the nilradical of a proper parabolic subgroup of Go. n

ANNALES SCIENTIFIQUES DE L’6COLE NORMALE SUPeRIEURE


PROPOSITION 2.17. - Assume that G is cuspidal and let 7r be an irreducible admissible representation of G. Then r is supercuspidal if and only if r is J-supercuspidal.

Proof. - This is an immediate consequence of Lemmas 2.15 and 2.16 and the corresponding result in the connected case.

We now drop the assumption that G is cuspidal. Let P = MN be a parabolic subgroup of G and let (T be an admissible representation of M. Then we let Indg(a) denote the representation of G by left translations on

7-L = {f E C”(G,V) : f(gmn) = s,~(nL)o(m)-‘~(g),~g E G,m E M,n E N}.

Here Sp denotes the modular function of P.

THEOREM 2.18. - Let r be an irreducible admissible representation of G. Then there are a cuspidal parabolic subgroup P = MN of G and an irreducible supercuspidal representation a of M such that 7r is a subrepresentation of Inds(o).

REMARK 2.19. - We will see in Corollary 3.2 that the group A4 and supercuspidal representation g in Theorem 2.18 are unique up to conjugacy.

Proof - Let p be an irreducible constituent of the restriction of 7r to Go. Then 7-r c Ind$ (p). Since p is admissible, there are a parabolic subgroup P” = M”N of Go and an irreducible supercuspidal representation r of M” such that p C Ind$ (r). Thus

T c Ind$(p) c Ind$(Ind$(7)) 2 Indgo(T).

Let P = MN be the unique cuspidal parabolic subgroup of G lying over P”. Let CT be an irreducible admissible representation of M such that r is contained in the restriction of (T to MO. By Lemma 2.16, g is supercuspidal. By Lemma 2.13 applied to M and M” we have

where Y is the group of unitary characters of M/&lo. Since r is contained in the restriction of P @ n to M” for any 71, all the representations 0 @ q are supercuspidal. Now

n c IndG,o (T) N s c Indg(a @ ~1).

Thus 7r c Indg(a @ r]) for some rl. n Let A(G) = &A( x w ) h ere 7r runs over the set of all admissible representations of G.

Similarly we have A(G”) and because of Lemma 2.11 it is clear that if f E A(G), then flGo E A(GO). Define the subspace Ar(Go) c A(GO) as in [13, $4.51. It is the set of functions in A(G’) which satisfy the weak inequality. Define

AT(G) = {f E A(G) : Z(x)flp E AT(G’) for all x E G}

where Z(x) f denotes the left translate of f by z. In other words, AT(G) is the set of functions in A(G) which satisfy the weak inequality on every connected component of G.

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If 7r is an admissible representation of G, we say 7r is tempered if A(n) c dT( G). The following lemma is easy to prove using the properties of matrix coefficients of 7r and 7r0 from the proof of Lemma 2.16.

LEMMA 2.20. - Let r be an irreducible admissible representation of G. Then r is tempered if and only if ~0 is tempered if and only if any irreducible constituent of ~0 is tempered.

Let x be an irreducible unitary representation of G and let 2 be the split component of G. We say that rr is discrete series if d(n) c L’(G/Z). Every unitary supercuspidal representation is discrete series since its matrix coefficients are compactly supported modulo 2.

LEMMA 2.21. - rf G is not cuspidal, then G has no discrete series representations. If G is cuspidal, then r is discrete series if and only if ~0 is discrete series if and only if any irreducible constituent of ~0 is discrete series.

Proof. - Suppose that G is not cuspidal. Then the split component Z of G is a proper subgroup of the split component Za of Go. Let K be any irreducible unitary representation of G. Then there is an irreducible unitary representation rl of Go so that 7r is contained in Ind& (~1). Thus as in the proof of Lemma 2.16, for any matrix coefficient f of 7t and any g E G, we have matrix coefficients fz of $, z E G/Go, so that for all go E Go,

fkwo) = c fz(go>. xEG/G“

Let w be the Zo-character of ~1. Then for any z E Za, go E Go, we have

fhd = c f.dSo~) = c w”(z)fz(go). xeG/GO &G/GO

Thus z ++ f(ggOz), z E Za, is a finite linear combination of unitary characters of Zo, and cannot be square-integrable on Za/Z unless it is zero. Now if f is square-integrable on G/Z, then go H f(gga) is square-integrable on Go/Z for all coset representatives g E G/Go, and z H f(ggaz) must be square-integrable on Zo/Z for almost all go, so that f(ggaz) must be zero for almost all go, z, and f = 0.

Suppose that G is cuspidal. Let ?r be a discrete series representation of G. Let f. be a matrix coefficient of ~0. Then there is a matrix coefficient f of 7r so that f. is the restriction of f to Go. Since f is square-integrable on G/Z, certainly fa is square-integrable on Go/Z.

Conversely, suppose that ~0 is discrete series. Let x1 be an irreducible constituent of ~0. For any matrix coefficient f of 7r we have

s,,, lf(g)124m = c / If(gg0)12d(goZ). geG/GO ’ Go/Z

As above, for fixed g E G, we have matrix coefficients f3: of I$,Z E G/Go, so that for all go E Go,

fho) = 1 f3cblo) sEG/GO

ANNALES SCIENTIF’IQUES DE L’hCOLE NORMALE SUPfiRIEURE

108 D.GoLDBERG AND R.HERB

Thus

since every fZ is square-integrable on Go/Z. Thus f is square-integrable on G/Z. n The following theorem can be proven in the same way as Theorem 2.18 using

Lemmas 2.20 and 2.21.

THEOREM 2.22. - Let x be an irreducible tempered representation of G. Then there are a cuspidal parabolic subgroup P = MN of G and an irreducible discrete series representation o of M such that r is a subrepresentation of Indz(a).

REMARK 2.23. - We will see in Corollary 3.2 that the group M and discrete series representation cr in Theorem 2.22 are unique up to conjugacy.

3. Intertwining operators

Let G be a reductive F-group with G/Go finite and abelian as in $2. If rrr and 7r2 are admissible representations of G, we let J( rl, 7r2) denote the dimension of the space of all intertwining operators from 7r2 to rl. We want to prove the following theorem.

Let (Pi, Ai), i = 1,2, be cuspidal parabolic pairs of G with Pi = Mi Ni the corresponding Levi decompositions. Let ai be an irreducible admissible representation of Mi on a vector space I,$:,, and let ri = Indg*(ai), where we use normalized induction as in $2. Let W = W(AlIA2) denote the set of mappings s : AZ + Al such that there exists yS E G such that S(Q) = ysa2y;l for all a2 E AZ.

THEOREM 3.1. - Assume that (~1 and u2 are either both supercuspidal representations, or both discrete series representations. Zf Al and A2 are not conjugate, then J(rl, ~2) = 0. Assume Al and A2 are conjugate. Then

Jhn) 5 c J(w$). SEW

COROLLARY 3.2. - Let o1 and ~72 be as above. Then J(nl, ~2) = 0 unless there is y E G such that MI = M;, o1 N o;.

COROLLARY 3.3. - Let u1 and ~72 be irreducible discrete series representations, and suppose that ~1 and R:! have an irreducible constituent in common. Then ~1 N 7r2.

Proof. - In this case J(nl, 7r2) # 0 so by Corollary 3.2 there is y E G such that Ml = M;,cq N o;. Thus

r1 = Indz,,, (al) = Indzpl (~7;) N IndzzN;(g2)

where N’ = NY 2 1 -I. Now P2 = M2N2 and Pi = M2Ni are two cuspidal parabolic subgroups of G with the same Levi component M2. We will see in Corollary 5.9 that there is an equivalence R( Pi : P2 : c2) between 7r2 = Inds2 (gz) and ~4 = Ind$ (~2). Thus

7r1211r;Yr2. n

4e StiRI! - TOME 30 - 1997 - No 1


Let W(C,) = {s E W(A1,Al) : (T: II pi}. We say g1 is unramified if W(cri) = (1).

COROLLARY 3.4. - Assume that CT~ is discrete series. Then J(r~,rl) 5 [W(ol)]. In particular, if CT~ is unramified, then ~1 is irreducible.

In order to prove Theorem 3.1 for discrete series representations we will need the following results about dual exponents. Let (r, V) be an irreducible representation of G and let V’ denote the algebraic dual of V. For z E G, let T(X)” : V’ + V’ denote the transpose of K(X). Let (P, A) be a cuspidal parabolic pair in G. Then a quasi-character x of A is called a dual exponent of r with respect to (P, A) if there is a nonzero I$ E V’ such that for all E E N, a E A,

(*I 7r(?t)“q5 = d, and K(u)~c$ = 6$(a)x(a.)$.

We will write Ym(P, A) for the set of all dual exponents of r with respect to (P, A). Let V = ‘& K be the decomposition of V into Go invariant subspaces and let rri be

the irreducible representation of Go on Vi, 1 5 i 5 k.

LEMMA 3.5. - Let (P,A) b e a cuspidal parabolic pair in G. Then

YK(P, A) = u:=~Y&‘~, A).

Proof. - Let x E Y, (P, A) and let 4 E V’ be a nonzero functional satisfying (*). Then there is 1 5 i 5 k such that &, the restriction of 4 to K, is nonzero. Now for any wi E Vi and go E Go we have

Now since 3 and A are both contained in Go it is easy to check that

(**) < rri(E)t+i,vi >=< &, vi > and < ~;(a)~$i, wi >= &((a)x(a) < C#Q,ZJ~ >

for all w; E V;,E E r, a E A. Thus x E Yrri(Po, A). Now assume that x E Y,, (PO, A) for some 1 5 i 5 Fc and let CJ$ E v be a nonzero

functional satisfying (**). Now rr is contained in the induced representation Ind$(?ri) so we can realize r on a subspace V of

‘H = {f : G -+ K : f(ggo) = dgo)-‘f(g),g E G,go E Go}

with the action of r given by left translation on the functions. Now define 4 E V’ by

< 44.f >=< q&if(l) >,f E v. Then C$ # 0 and it is easy to check that for all go E Go, f E V, we have

From this it easily follows that since $i satisfies (**), C$ satisfies (*). Thus x E Xr(P, A). n

ANNALES SCIENTIFIQUES DE L’kOLE NORMALE SUPlkIELJRE


LEMMA 3.6. - Assuye that G is cuspidal and let 7r be a discrete series representation of G. Then Y, (P, A) n A = 0 f or every cuspidal parabolic pair (P, A) # (G, 2).

Proof. - This follows easily from Lemma 2.21, Lemma 3.5, and the corresponding result for connected groups. n

Let (P,“, Ao) be a minimal p-pair in Go and let PO be the cuspidal parabolic subgroup lying over Pi. We will call (PO, Ao) a minimal parabolic pair in G.

LEMMA 3.7. - Let (P, A) b e any parabolic pair in G. Then there is x E Go such that Pa c ~Px-~,xAz-~ c AO.

Proof. - Let M = CG(A) and let P” = P fl Go, M” = M n Go. Let A’ be the split component of MO. Thus A c A’. Now (PO, A’) is a p-pair in Go so there is z E Go such that Pi c zP”z-l and zArc-’ c xA’~-l c Ao. Now MO = t&(AO) c c~(zAx:-l) = XMX-l so that PO = MOP,0 c (xM~-‘)(zP~z-~) = ZPZ-l. n

We will say a parabolic pair (P, A) is standard with respect to the minimal parabolic pair (PO, Ao) if PO c P and A c Ao. We say (P, A) is semi-standard with respect to (Po,Ao) if A c Ao.

Fix a minimal parabolic pair (PO, Ao) of G. Let NG ( PO, Ao) denote the set of all elements of G which normalize both PO and A,+ Write k&(Po,Ao) = NG(Po,A~)/M~. If (P,A) is any parabolic pair of G which is standard with respect to (PO, Ao) and A/f = CG(A), write Nnn(Po, Ao) = M n NG(PO, Ao) and W*l(Po, Ao) = NIM(Po, Ao)/Mi.

LEMMA 3.8. - We can write P as a disjoint union

P=U WEWhf(PO,Ao)wpo.

Proof - We first prove the result when P = G. Let y E G. Then (yP:y-‘, yAoy-l) is a minimal parabolic pair in Go and hence is conjugate to (P,“, Ao) via an element of Go. Thus there is g E Go such that gg normalizes both A0 and Pi. But then gy also normalizes Ma = CG(A~) and PO = MOP:. Thus gy E NG(Po,Ao).

Thus the coset gG” has a representative 72 E NG (PO, Ao) which depends only on the coset w of n in WG(P~,A~). Since NG(Po,A~) n Go = M$‘, n,i,n~~p E Na(Po,Ao) determine the same coset of Go in G just in case they are in the same coset in WG( PO, A,). Thus we have the disjoint union

G=u w~wc(~o,~,)wG~.

Now let (P, A) be arbitrary. There is a subset 0 of the simple roots A of (P,“, Ao) corresponding to P” so that P” = Pi. Note that WG( PO, Ao) acts on A and that w-lP”w = PO 0 wO. Let w E IVG(P~, Ao) and suppose that P n wG” # 0. Let wgo E P n wG”,go E G O. Then wgo normalizes P” so that w-l Pow = goP”gol is conjugate to P” in Go. But w-lP”w = PzB is a standard parabolic subgroup of Go and so PL = gap O -’ = PO. Hence go E NQ (P”) = PO, WB = 8, and w has a representative 90 in P. Hence UI normalizes A = {u E A0 : Q(U) = 1 for all LY E 6)). But Na(A) n P = M. Thus w E I&(Po,Ao). n

In order to prove Theorem 3.1 we extend the proof of Silberger in [13, $2.51 to the disconnected case. Fortunately many of the technical results on intertwining forms needed

4e S6RIE - TOME 30 - 1997 - No 1

SOMERESULTS ON THE ADMISSIBLERBPRESENTATIONS 111

are proven in [13, $11 for any totally disconnected group and so can be directly used in our case. We follow Silberger’s notation. Let (PI, A,) and (Ps, AZ) be cuspidal parabolic pairs in G. We may as well assume that they are standard with respect to a fixed minimal parabolic pair (Pa, A,).

We need to study the orbits for the action of Pr x P2 on G given by y. (pr , ~2) = pr’yp2. Recall that Go = U,P~wP~ where w runs over W(G’,Ao) = NQ(AO)/M~. Thus, G = u,,wG’ = U,,,wP~vP~ = U,,,P~wvP~ where w E WG(Po,Ao),v E W(G’,Ao). Now since Pi c Pi, i = 1,2, each double coset P,yP, can be represented by y = WV, w E WG(P~,A~),V E W(G”,Ao). Write Wi = WM,(Po,AO),i = 1,2.

LEMMA 3.9. - Let c3 = P,wovP2 be an orbit of PI x PZ @ G where wg E w~(Po,Ao), w E W(G’; Ao). Then for UI E WG(PO, Ao), 0 n wG” is empty unless w E W,woW2. If w E WIwoW,, then c3 f’ wG” = ~0, where 0, is a finite union of orbits of (w-l PFw) x Pi in Go, all of which have fixed dimension do equal to the dimension of the orbit 00 = (w~‘P~wo)vP~.

Proof. - Using Lemma 3.8 we can write

PlWOVP2 = uw,,,, W~P~WOVW~P;

where wi E Wi = WM,(PO, A,), i = 1,2. But

W&~WOVW~P~ = w(w-1P~w)w;1Yw2PLp

where w = W~WOWZ E w~(Po,Ao). Thus w~P~wovw~P~ c W~WOUJ~G~ and for fixed w E WlWOW2,

P~wOVP:! n wG” = w u,, (~-~Pj’w)w;~vw~P;

where w2 runs over elements of W2 such that ww;l w;’ E WI. Finally,

(w-1P$i1)w~1uw2P~ = w2 -1(w;1P;w&lP;w2

so that

dim(w-lP~w)w~lvwsP~ = dim(w;‘P~wa)vP,” = do.

n Because of Lemma 3.9 each orbit 0 has a well-defined dimension do. For each integer

v 2 0, let O(Y) denote the union of all orbits of dimension less than or equal to V. Set 0(-l) = 0.

LEMMA 3.10. - For each v 2 0, O(Y) is a closed set in the p-adic topology. Further, if 0 is an orbit of dimension d, then 0 U O(d - 1) . 1s a so 1 closed in the p-adic topology.

Proof. - Using Lemma 3.9 we see that 0(v) n wG” = we),(v) where 0, (v) is the union of all orbits of (w-‘P,“w) x Pt in Go having dimension less than or equal to V. This set is closed in the p-adic topology by [13, pg. 931. Thus O(Y) is

ANNALES SCIENTIFIQLJES DE L’kCOLE NORMALE SUPi?RlELJRE


a finite union of closed sets, hence is closed. Similarly, if 0 has dimension d, then [O U O(d - l)] n wG” = ~(0, U O,(d - 1)) where 0, U O,(d - 1) is a finite union of sets of the form 0’ U O,(d - 1) where 0’ is an orbit of ( w-‘P~w) x Pi in Go of dimension d. These are also closed by [13, page 941. n

Let E = VI @ V, where I$ is the space on which ~‘i acts, i = 1,2. Let 7 be the space of all E-distributions To on G such that

T0(%MP2)4 = ~0(~1(Pl)~~2(P2)~~l(P1)-1 8 a2(p2)-la) for all (pi, ~2) E Pi x Pz, (Y E C,-(G : E). Here A and p denote left and right translations respectively and & is the modular function of Pi, i = 1,2. The first step in the proof of Theorem 3.1 is the inequality [13, 1.9.41

(3.1) I(TI, ~2) 2 dim 7.

Here 1(ri, 7ra) is the dimension of the space of “intertwining forms” defined in [ 13, 5 1.61. It is related to the dimension of the space of intertwining operators by 1(~i, ~2) = J(771, nz) where iii is the contragredient of rl [13, 1.6.21.

If 0 is an orbit of dimension d, write I(0) for the vector space of To E 7 with support in 0 U O(d - 1) and 7V for the space of those with support in O(Y). We have

7=X7(0) u

and

(3.2) dim 7 5 c dim(7(0)/‘&(op-l). 0

LEMMA 3.11. - Suppose that (PI, Al) and (P2, A ) 2 are semi-standard cuspidal parabolic pairsinG. ThenMlnP2 = (M,nM,)(M,nN,) = *P 1 is a cuspidal parabolic subgroup of iI41 with split component *Al = AIAZ.

Proof. - We know from the connected case [13, p. 941 that *PF = MF n Pi is a parabolic subgroup of MF with split component *Al = AlA and Levi decomposition *Pf = (M,OnM,O)(M,OnN,). Thus th ere is a cuspidal parabolic subgroup *PI of Mi with split component *Al and Levi decomposition *PI = *MI *NI . Here *Mi = CM, ( *Al) = CMl(A1A2) = MI n C&AZ) = MI n Ad2 and *IV1 = MF n A$. = Ml n N2. Clearly *PI = (MI n M2)(MI n N2) c Ml n P2. Thus we need only show that Mi n P2 c *PI.

Let II: E Mi n P2. Using the Levi decomposition of P2 we can write x = m2n2 where -1 -1 m2 E M2,n2 E N2. Since m2n2 E Ml = CG(A~) we have m2n2ain2 m2 = al for

any al E Al. This implies that a11n2a1ng1 = aT1m;1a1m2. But since (P,,Al) and (Pz,Ag) are semi-standard, we have Al c A0 c Cc(A2) = A&J. Thus a;‘n2aln;l E NZ and a;‘m;‘airn2 E Ms. Hence aT1n2aing1 = al’rn;‘airns E N2 n A42 = (1). Thus n2 and 1~2 both commute with al so that 7~2 E Mi n N2 and m2 E MI n A&. n

Return to the assumption that (PI, A,) and (P2, AZ) are standard with respect to (PO, A,) and that y E W(A 0 ) so that (P!, A;) is semi-standard. Using Lemma 3.11 we know that

de SliRE - TOME 30 - 1997 - No 1


*PI = MI n Pz and *Pz = MZ n Py-’ are cuspidal parabolic subgroups of MI and M2 respectively.

Let E’(y) denote the space of linear functionals 4 on E = VI @ VZ such that

sl(p)~s2(py-y < qh(p)w @aa2(py-1)Z12 >= bl.,;(P> < d$Wc3~2 >

for all p E PI n Pl, wi E V,,i = 1,2.

LEMMA 3.12. - Let m E *MI = MI n M,Y. Then

Splnp;(m) = 6,1(m)~S,2(my~~)fS~(m)~~2(my~1)3

where & denotes the modularfunction for *P; and & the modularfunction for Pi, i = 1,2.

proof. - Define the homomorphism S : *MI -+ R+ by

6(m) = 6*1(m)~~*2(mY-1)~S~(m)~d2(my-1)~S~~~~~(m)-1,m E *MI.

By [13, $25.21, the restriction of S to *MI n Go is trivial. Now since the quotient of *MI by *MI n Go is a finite group, 6 = 1 on all of *Ml. n

The proof of the following lemma is the same as that of Lemma 2.5.1 in [ 131.

LEMMA 3.13. - ZfqS E E’(y), then q5 vanishes on VI( *PI) @I VZ + VI @ VZ( *P2). Let m E *Ml, w1 63 v2 E E, and 4 E E’(y). Then

< 4, al(m)m @ a2(mY -l)‘u:! >= S,l(m)+S*2(mY-1)+ < 4,711 @ 212 > .

COROLLARY 3.14. - Zf Al = A;, then

< 4, ul(m)wl 8 02(mye1)v2 >=< 4, w1 C3 212 >

for ah m E *M~,Q @ 212 E E, and 4 E E’(y).

Proof. - As in [13, $2.541, AI = Ai implies that *IV, = *IV2 = (1) so that *PI and * Pz are reductive and S,r = S*2 = 1. n

COROLLARY 3.15. - Assume that 01 and ~2 are either both supercuspidal or both discrete series. Then E’(y) # (0) implies that Al = AZ.

Proof. -This follows as in [13, §§2.5.3,2.5.5] using Proposition 2.17 and Lemma 3.6. W The following Lemma is now proven as in [ 13, $2.5.71.

LEMMA 3.16. - Assume that a1 and 02 are either both supercuspidal or both discrete series. Let 0 be an orbit in G of dimension d. Then dim 7(0)/&l = 0 unless there exists y E 0 such that AI = A;. If AI = Ai for some y E (3, then

dim(7(O)/Ll) I I(al, 4.

Proof of Theorem 3.1. - Combine equations (3.1) and (3.2) to obtain

I(m, 7~) 5 dim 7 5 c dim(7(O)/Ti(o)-l).

Cl

ANNALES SCIENTIFIQUES DEL'l?COLENORMALESUPtiRIEURE


Now using Lemma 3.16, if Al and A2 are not conjugate, dim(7(0)/7d(o)-l) = 0 for every orbit 0 so that 1(7r1, ~2) = 0. If Al and A2 are conjugate we have, again using Lemma 3.16,

To complete the proof note that J(rl,xz) = 1(7ii, xz), J(oi, 0:“) = I(c?~, a;*) and iii = Indgl (ai). Thus the statement with dimensions of spaces of intertwining operators rather than intertwining forms follows by substituting iii for TV. n

4. R-groups for Ind$(cTa)

In this section we will study representations of G which are induced from a parabolic subgroup P” = M”N of Go. Because in this section we will only work with parabolic subgroups of Go, we will simplify notation by dropping the superscripts on P” and MO.

Let P = MN be a parabolic subgroup of Go and fix an irreducible discrete series representation (a, V) of M. Define

-4 T-lp(cr) = {f E C”(G, V) : f(zmn) = 6, (m)a(m)-‘f(z)

for all z E G, m E M, r~ E N}

where SP is the modular function on P. Then G acts by left translation on Xp(a) and we call this induced representation Ip(g). We will also need to consider

IFlOp(a) = (4 E C”(GO, V) : +(zmn) = s,“(m)g(m)-‘4(x)

for all z E Go, m E M, 71 E N} .

Go acts by left translation on ‘H:(a) and we call this induced representation IF(g). It is well known that the equivalence class of Ii(a) = Indg’(a) is independent of P. But

Go IP(~) = Indg(cT) N Ind$Indp (u),

so that the equivalence class of Ip(c) is also independent of P. We denote the equivalence classes of Ip(a) and 1,(a) by i~,~(cr) and ‘~Gu,A~(o) respectively. If rr is a representation of Go we will also write i~,co(r) for the equivalence class of the induced representation Indzo (7r).

We first want to compare the dimensions of the intertwining algebras of i~,hr(g.) and ico,~(a) . For this we need the results of Gelbart and Knapp summarized in $2 and the following facts.

LEMMA 4.1. - Suppose ~1 and r2 are irreducible representations of Go. Then ic,~o (~1) and ~G,GO (~2) have an irreducible constituent in common if and only if 7r2 N $ for some g E G. In this case they are equivalent.

4' StiRiE - TOME 30 - 1997 - No 1


Proof. - For i = 1,2, write

iG,&ri) = Ti c nI; @ x x~XlX(K)

as in Lemma 2.13. Suppose that ~G,GO(K~) and i~,Go(Ka) have an irreducible constituent in common. Then III @ x1 N IIs @ x2 for some ~1, xz E X. Now

Thus 7r2 11 7ry for some g E G. Conversely, if ~2 N 7ry for some g E G, then clearly

iG,G+2) = iG,G’+;) 2 iG,G” (rl). n

LEMMA 4.2. - Suppose that for some g E G, both n and rg are irreducible constituents ofiGO,M(u). Then there is 20 E Go such that gzo E NG(o) = {z E NG(kf) : o” N o}. Conversely, if n is an irreducible constituent of i~o,~(o) and if g E NG(u)GO, then rg is also an irreducible constituent of i~o ,M (u) and r and rg occur with the same multiplicities.

Proof. - Suppose that 7r, x g C iGO,M(u). Then since 71 ’ c iGO,Ms(u’), we see that iGO,M(u) and iG0,~s (ag) have an irreducible constituent in common. Thus there is x0 E Go such that MgzO = A4 and P0 cv g, i.e. gso E NG(o). Conversely, if g E NG (a)GO, then the multiplicity of 7r in i~o ,M (c) is equal to the multiplicity of +J in iGO,M(~)g ‘v iGO,,&+g) 11 iQ$,f(~). n

LEMMA 4.3. - Let x be an irreducible constituent of EGO ,~(o) and let G, = {x E G : IT” N 7r). Then

[Gx/Gol = [NG,(~)INGo(~)I

where NGO((T) = NG(~) tl Go and NG,(~) = NG(~) n G,. Proof. - Consider the mapping from NG, (g) to G,/G” given by g H gG”. Its kernel

is NG,(cr) n Go = NGO(U). Further, given g E G,, ~9 N 7r occurs in ~Go,M(u), so by Lemma 4.1 there is x0 E Go such that gxa E NG(~). But ngzO N rrg N r so that gxa E G, n NG(~) = NG, (a) and gxaG” = gG”. Thus the mapping is surjective. n

Let C(O) denote the algebra of G-intertwining operators for ‘LG,M (0) and let C”(o) denote the algebra of GO-intertwining operators for iG0,~(u).

LEMMA 4.4. - dimC(a) = dimC”(~)[Nc(a)/Nco(a)].

Proof. - Let

iGo,&&) = c ‘%rr 7&9(O)

be the decomposition of iGO,M(u) into irreducible constituents. For rl, r2 E S(U), we will say that w1 N 7ra if there is g E G such that 7r2 11 ry. Then using Lemma 4.2, we have ~1 - 7r2 if and only if ‘ITZ N 7$’ for some g E NG(~). We can write

iGO,&,&) = c m, c X9. es(o)/- ~ENG(~)ING, Cm)

ANNALES SCIENTIFIQUES DE L’6COLE NORMALE SUP6RIEURE


Now

where II, is an irreducible representation of G such that rr c II, IGO. Because of Lemma 4.1, the representations IIT @ x are pairwise inequivalent as T ranges over S(G)/ N and x ranges over X/X(&). Thus

But by Lemmas 2.13 and 4.3,

r;[x/x(&)] = [G/G’] = [NG,(~)/NG+)]*

Thus

dimC(a) = c ~~[~G(~)I~G,(~)I~[~G,(~)I~Go(~>I TES(U)/N

= [NC+(~)/NGO(~)] C &[NG(cJ)/NG~(~)I 763(U)/N

= [NG(~)/NGO((T)] dim C’(O).

We now want to find a basis for C(U). We proceed as in the connected case. Let A be the split component of the center of M and write a& for the dual of its complex Lie algebra. Each v E a& determines a one-dimensional character xy of M which is defined by

xv(m) = q-+)++,rn E h/l.

We write (Ip(a : v), N~lp(a : v)) and (I;( o : ZI) ,7-$ (g : v)) for the induced representations of G and Go as above corresponding to u, = CJ 8 xv. Let K be a good maximal compact subgroup of Go with respect to a minimal parabolic pair (PO, Ao) of Go such that PO C P, A C Ao. Then Go = KP and we also have the usual compact realization of I:(a) on

7iE(a) = {f~ E C”(K,V) : f~~(kmn) = a-+n)f~(k) forallkEK,mEMnK,nENnK}.

The intertwining operators between F$( g, V) and ‘FI$ (a) are given by

F,K(v) : 7-&o : v) -+ T-$(u),

F,K(L+#o) = 4(k), q5 E %$(a : v), k E K.

4e S#iRIE - TOME 30 - 1997 - No 1


For all f~ E ‘Fl$(a),z E Go,

Here for any n: E Go, K(X) E K, p(z) E A4 are chosen so that ic E K(z)IJ(z)N. Since we don’t know whether there is a “good” maximal compact subgroup for G, we

don’t have a single compact realization for Ip(c). However we can proceed one coset at a time as follows. Write G as a disjoint union of cosets, G = UfLE=,xiGo. For any f E XFlp(a, v), 1 5 i < Ic, we can define

f(x), .fiCx) = { o

if x E ziGo; 1 otherwise.

Then fi E XFlp(a : V) for each i and f = C,“=, fi. For each 1 < i < k, define

F;(v) : Xp(a : v) -+ T-lpK(cJ)

Define

F$(v)f(k) = f(Xik), f E T-lp((T : v), k E K.

&J(qffK(x) = C ~~~(~(xo))~;‘(~(xo))~K(K(xo)), if II: = X~XO,XO E Go;

0, otherwise.

Then F~(v)F$(v)-‘~K = fK for all fK E X$(O) and Fb(~)-lFb(~)f = fi for all f E ‘Flp(o : v).

If P = MN and P’ = MN are two parabolic subgroups of Go with Levi component M, then we have the formal intertwining operators

JO(P’ : P : CT : v) : 7fgo : v) + ‘FI&(cT : 2-J)

given by the standard integral formula

JO(P’ : P : c7 : 2+$5(x) = .I

NnN, $(xn)dz, x E Go.

Here MN is the opposite parabolic to P and &X is normalized Haar measure on n rl N’. We can define

J(P’ : P : (T : v) : T-lp(a : v) + IFlp(a : v)

by the same formula

J(P’ : P : (7 : v)f(x) = .I

Ivnlv, f(x+c x E G.

ANNALES SCIENTIFIQUES DE L%COLE NORMALE SUPBRIEURE


In order to talk rigorously about holomorphicity and analytic continuation of these operators we transfer them to the compact realizations. Thus we have

J$(P’ : P : o : u) : ?-L:(a) + 7-$(o)

defined for j’~ E Xc(a),k E K, by

J::(P’ : P : u : V)&(k) = Ffi(V)JO(P’ : P : 0 : t&qo)-lf&k)

rz J’ ’

Ivn~ hp~(~(k.Si))u~l(~(kA))f~(~(~~))~.

THEOREM 4.5 (Harish-Chandra [7, 131). - Suppose a is an irreducible discrete series representation of M. There is a chamber CL& (P’ : P) in & such that for u E L& (P’ : P) the formal intertwining operator J”( P’ : P : u : u) converges and defines a bounded operator. For a fixed fu E 7-l:(a), the mapping

from &(P’ : P) to 7-tg( ) h 1 o IS o omorphic. Further, it extends to a meromorphic function on all of a&.

Because of Harish-Chandra’s theorem we can define J”(P’ : P : a : V) for all Y E r& by

J”(P’ : P : a : V) = F;(V)-’ J;(P’ : P : o : u)F,K(u).

COROLLARY 4.6. - Suppose o is an irreducible discrete series representation of M. Then the formal intertwining operator J(P’ : P : o : V) converges and defines a bounded operator for v E a& (P’ : P). Further, for all v E &(P’ : P), we have

J(P’ : P : u : u) = -& F;,(u)-‘J$(P’ : P : u : u)t$(u). i=l

Proof - It is clear from the definitions that for v E a& (P’ : P), f E Xp(a : v), x E G, x0 E Go,

J(P’ : P : o : u)f(zxo) = J”(P’ : P : o : v)q5(zo)

where 4 = I(z-‘)fl GO is the restriction to Go of the left translate of f by 5-l. Thus the integral converges. Fix 1 5 i < k. An elementary calculation shows that

F;,(u)J(P’ : P : o : v)F;(v)-~ = J;(P’ : P : o : u).

Thus for any f E tip(a), since clearly J(P’ : P : o : u)(fi) = (J(P’ : P : o : u)f)i we have

F&(v)-~J;(P’ : P : CT : v)F;(v)f = F;,(v)-lF$,(~)J(P’ : P : u : ~)F~(u)-~F~(u)q3 = (J(P’ : P : CJ : ~)f)~. n

4e S6RIE - TOME 30 - 1997 - No 1

SOME RESULTS ONTHE ADMISSIBLE REPRESENTATIONS 119

Because of Corollary 4.6 we can define J(P’ : P : (T : V) for all v E & by the formula

J(P’ : P : o : V) = &(V)-‘J:,(P’ : P : CT : zqqu). i=l

COROLLARY 4.7. - Let Y E a&, f E ‘Bp(a : V),X E G,XO E Go. Then

J(P’ : P : a : v)f(zzo) = JO(P’ : P : o : V)$(Xg)

where $ = 1(x-‘)~(Go.

Proof. - Suppose that 1 5 j < k. Because J”( P’ : P : (T : V) commutes with the left action of Go, it is enough to prove the result for 2 = xj. For any

u E a;, f E ?-lp(o : u),xo E Go, J(P’ : P : a : v)f(xjxo)

= 2 F;,(V)-lJ;(P’ : P : (T : v)F~(v)f(~jzO) i=l

= F;,(Y)-lJ$(P’ : P : fJ : v)F$(v)f(xjzo)

= F$(V)-9;(P : P : (T : V)FpK(z+$j(XlJ)

where +j = L(~;‘)f]~0. n Fix scalar normalizing factors r(P’ : P : 0 : V) as in [2] used to define the normalized

intertwining operators

RO(P’ : P : g) = r(P’ : P : c7 : 0)-‘bO(P’ : P : I7 : 0).

Using Corollary 4.7, the fact that r(P’ : P : (T : v)-lJ’(P’ : P : (T : V) is holomorphic and non-zero at v = 0 will imply that r(P’ : P : 0 : v)-‘J(P’ : P : (T : V) is also holomorphic and non-zero at v = 0. Thus we can use the same normalizing factors to define

R(P’ : P : (T) = T(P’ : P : 0 : 0)-lJ(P’ : P : 0 : 0).

For the normalized intertwining operators we will also have the formula

R(P’ : P : o)f(zxo) = RO(P’ : P : a)#&)

where notation is as in Corollary 4.7.

LEMMA 4.8. - Suppose PI, P2, and P3 are parabolic subgroups of Go with Levi component IVl. Then R(Pl : P3 : a) = R(Pl : P2 : (T) R( PZ : P3 : a).

Proof. - This follows easily using Corollary 4.7 from the corresponding formula for the connected case. n

LEMMA 4.9. - Suppose PI and P2 are parabolic subgroups of Go with Levi component n/l. Then R(P, : PI : a) is an equivalence from ‘Flp, (~7) onto XFlp, (a).

ANNALES SCIENTIFIQUES DE L%C!OLE NORMALE SUP6RIELJRE


Proof. - It follows from Lemma 4.8 that

R(P, : P* : a)R(P2 : PI : 0) = R(P1 : PI : f7).

But it follows from the integral formula that J(Pl : PI : g : V) is the identity operator for v in the region of convergence, and hence for all V. Thus R(Pl : PI : a) is a non-zero constant times the identity operator and so R(P2 : PI : C) is invertible. n

Let z E NG(M). Then if PI = MN1 is a parabolic subgroup of Go with Levi component M, so is Pz = XPX-’ = xMx-~xN~x-~ = MN,. Let dnl and dn2 denote normalized Haar measure on Nr and N2 respectively. (Thus dni assigns measure one to K n Ni, i = 1,2.) Define oP, (x) E R+ by

For all m E M,

Further, if x E NG (K) n NG (M), then ctp, (x) = 1. The following lemma is an easy consequence of the definition.

LEMMA 4.10. - Let x,y E NG(M). Then

Moreover if m E M, y E NG(M), then

wJ(ym) = (3P(g)sP(7rL).

Now let No = {g E NG(M) : erg N U} and let We = NG(~)/M. Let WGo(~) = (NG(a) n G”)/M. If UI E We, g can be extended to a representation of the group Mw generated by M and any representative n, for w in NG (a). Fix such an extension and denote it by gW. Now we can define an intertwining operator

Ap(w) : Flu,-lr,(a) + x~lp(g)

The a,-~~, term is not used in the connected case because coset representatives n, can be chosen in K where ~~-1 pW = 1. In the general case we don’t know if there is a natural choice of coset representatives with o,-lp, = 1. Thus we add the a,-lp, term so that Ap( w) is independent of the coset representative n,, for w. Ap(w) does however depend on the choice of the extension crW.

4e SBRIE - TOME 30 - 1997 - No 1


LEMMA 4.11. - The intertwining operator Ap(w) is independent of the choice of coset representative n, for w E WG(g). For WI, w2 E WG(U) there is a non-zero complex constant cp(w1, ~2) so that

AP(WIW~) = cp(w1, w~)A~(wI)A,;I~,, (~2).

Proof - Using Lemma 4.10 we have for any m E M,f E ‘&-1~~(g),

Thus the intertwining operator is independent of the choice of coset representative. Let wl, w2 E WG(o) and fix coset representatives 7x1, n2 for wl, ws respectively. Then

we can use n1n2 as a coset representative for 201~2. We will also write PI = w,lPw~ and PI2 = w;‘w;’ Pw1w2. For any w E WG(c) and representative n, for w we have

f3w(n,)c7(m)~,(n,)-1 = a(n,mn,l)

for all m E M. Thus

uwu11w2(nln2)flw2 (n2)-1~wl (nJ1

is a non-zero self-intertwining operator for cr and hence there is a non-zero constant c so that

~wlwz(nln2) = c~,,(nl>h(n2).

Further, from Lemma 4.10 we have

By definition, for any z E G, f E X~lp,, (u),

&4~1~2)f(~) = ~2u1wz (nln2)~p,,(n1n2)3f(Zn1n2)

= ca,,(nl)ap,(7L1)3uzU,(n2)Qlp,,(n2)3f(2n1n2)

= CAP h)A,;+,, (wz)f(4.

If w E WQ (c) and n, is chosen to be in K, then we also have

A’(w) : 7-&,,,(a) + 7-L;@)

given by

(A’(wM)(z) = aw(nw>4(~nw).

ANNALES SCIENTIF'IQLJES DE L'kOLE NORMALE SIJP6RIEURE


The compositions

R’(wo, a) = AO(w~)Ro(u~,lPwo : P : a), w. E WGo(c),

R(w, a) = Ap(w)R(dPw : P : a), w E WG(g),

give self-intertwining operators for 1: (c) and IP (0) respectively.

LEMMA 4.12. - There is a cocycle 7 so that

R(WlW2r a) = 77( w , ,~2)R(W, cJ)R(w2, g)

for all wl,w2 E WG(O).

Proof. - Using Lemmas 4.8 and 4.11, we have

R(w1w2, a) = A~(w~w~)R(w,~w,~Pw~,~~ : P : cr)

= ~p(w1,.1li2)A~(,~l)Aul,l~u~~(.(~a)

x R(w,~w,~Pw~w~ : w,‘Pw2 : (~)R(w,lPw~ : P : CT).

We will show that there is a non-zero constant c’p( WI ! ~2) so that

4q’PWl (~w~)R(w;~u~ -+wl,wz : w,~Pw~ : c~)A~(uI$~

= cIp(wl, w2)R(w,lPw1 : P : cr).

When this is established we will have

R(ww2,g) = ~P(w~,w~)c~(w~,'uI~)AP(w~)

x R(w,‘PwI : P : ~r)Ap(w~)R(w,~Pw~ : P : 0)

= cp(wl, ~u,)c~~(~~~~,,uIz)R(w~, a)R(w2, c).

Thus 11 can be defined by q(wl,w2) = cp(wl, w2)c>(wlr wp). It is immediate from the formulas for the composition of the operators R(wi, P) that q~ is a 2-cocycle.

In order to prove the above identity we need to go back to the original definition of the standard intertwining operators. WG (g) acts on C& and ~~(n;~rnn,) = xw,,(m) if n, is any representative of w E WG(O). Now for any v E CZ~ , f E 7-L-1 pw (a! v), IL‘ E G, m E M, n E N, if Ap(w) is defined exactly as above, we have

Mw)f(xm4 =a,(n,)a,~~Pw(n,)ff(xmnn,)

=a~(nw)(W,-lp~,(n,)~S,~lp,(n,‘m-’n,)-3a n ‘rn-ln~)xv(nwlnL-In,)f(zn,) 1 =fi ( 1-I ( >- x ( )- 20 m 1

( > ff, - ( 1~~ n,

=6r(~)-b(rn)-1x::::(:)-1~(~)f(x).

Kc i 272, 1

Thus Ap(w) maps XFl,~lp,(a,v) to ‘Hp(g,‘wv). Now write PI = w,‘Pw~, PZ = wq1pw2, PI2 = ?U,~W,~ Pwlw2 and consider the composition

ApI (wz)J(P12 : P2 : o : w34Ap(w2)-’

4e S,?RIE - TOME 30 - 1997 - No 1


It maps

XP(cJ, v) -+ XP* (a, w2 -‘4 + ~P12(~, w;ly) --+ XP, (0, v).

If v is in the region of convergence for the integral formula for J(Pi:! : Pz : (T : w;‘v) and &z denotes normalized Haar measure on xZ n N12, then we have

Apl(w2)J(P12 : P2 : o : ~;~v)A~(w~)-‘f(x)

= (T,,(~~,,)Q~,,(~,,)~J(P~~ : P2 : (T : WZ~I/)A~(W~)-‘~(Z~L,,)

But Na n Nr, = w;’ (x n N i wg so if & denotes normalized Haar measure on F n N,, ) there is a positive real number T so that

for all 4 E Cr (N n Ni). Thus

Aph2)J(P,2 : P2 : o : ~34A~(w~)-~f(2~)

Setting

we have

A~l(w2)J(P12 : P2 : Q : w;‘~4Ap(w$~ = c;(wl, w~)J(P~ : P : CT : I/)

for all Y in the region of convergence for the integral formula for J(P,, : P2 : f~ : w;‘v). By analytic continuation, the identity is valid for all V. Now divide both sides of the equation by r(P12 : P2 : g : w;’ V) and evaluate at v = 0. We obtain

A9(w2)R(P~2 : P2 : +~P(wz) -1

= clp(w1, W2)R.(P~ : P : CT)

where cIp(w1, w2) = cS(w1, WZ)T(Pl : P : (T : O)T(Pl:! : P2 : CT : 0)-l. n

For 4 E 7$(c) define f = a($) E Xp(a) such that f(x) = 0 if 2 @’ Go and f(xo) = 4(x0) if xo E Go.

ANNALES SCIENTIFIQUES DE L’kOLE NORMALE SUPBRIEURE


LEMMA 4.13. - Let f = f@(4) as above. Then for all w, w1 E W,(o)? x0 E Go,

unless w = wlwo,wg E WGO(C). Zfw = W~WO,WO E Wp(o), then

Proof. - For any z E G, w E WG(~), using Corollary 4.7,

R(w-’ Pw : P : a)f(x) = R’(w-lPw : P : a)&(l)

where fl is the restriction to Go of Z(x-‘)f. Now since f = a($) is supported on Go, #=O unless 2 E Go. If x0 E Go, then R(w-lPw : P : a)f(zo) = R”(w-lPw : P : a)$(~~).

Now by definition,

R(w,a)f(zon,;) = ~~(~~)~w-lp~(~w)~R(w-lPw : P : o)f(z,n,;n,).

By the above this is zero unless n;in, E Go, that is unless w = wiwa, w. E Woo. In this case, using Lemma 4.12,

Rhwo, Mzon,;) = r1(wr wo)R(w : g>R(wo : Mzon,;) = v(w, wo)azo1 (G&,;+,, (~w,)3R(wPw~1 : P : o)R(wo : o)f(zo) = ~(w~rwo)~zu~(~,,)~,~~~~,(~,,)~Ro(~~P~;l : P : c)R’(wo : g)$(zo). n

Recall that if we write IV& (g) for the subgroup of elements w E WGO (0) such that R’(w,a) is scalar, then W$O(c) = W(+i) is generated by reflections in a set a1 of reduced roots of (G, A). Let @+ be the positive system of reduced roots of (G, A) determined by P and let +I + = al n fD+. If we define

Rt = {w E VVp(a) : wp E ip+ for all ,B E a:},

then IT’,0 (u) is the semidirect product of Rz and W( (ai). Rz is called the R-group for 1: (a) and the operators

{Rob-, 01, r E R:l

form a basis for the algebra of intertwining operators of I:(g). We will define

R, = {w E WC(~) : w@ E @+ for all ,b E (at}.

Clearly R, n Go = Rz.

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LEMMA 4.14. - R(w,(T) is scalar ifw E W(@I), and WG(~) is the semidirect product of W((a,) and R,.

Proof - Fix w E W(+,),n, E K a representative for w, and f E tip(g). Then for all 2 E G,

R(w, g)f(x) = a,(n,)R(w-lPw : P : a)f(zn,) = ~,(n,)R~(w-~Pw : P : o)$(nW)

= where $(x0) = f(zzc) for all ~0 E Go.

Thus Rh df(4 = Rob, Ml).

But since w E W(Ql) there is a constant c, such that RO(w,a)$ = c,$ for all C#I E 7$(a). Thus

R(w, ~)f(4 = cud(l) = cd(z)

so R(w, o) is scalar. We must show that for any w E WC (0)) war = ipr so that

R, = {w E W&) : w@ = a;}.

Then as in the connected case it will be clear that W G o is the semidirect product of ( ) R, and W(@,). Let w E WG((T),(Y E Qr, and let s, E W(+l) denote the reflection corresponding to CY. Then ws,w-r = s,, E WGO (P). But R(s, : O) is scalar so that using Lemma 4.12, so is R(s,,,a) = R(ws,w-l : c). This implies as above that R’(s,, , g) is scalar. Hence s,, E W(&) and wcx E @r. n

LEMMA 4.15. - The dimension of C(a) is equal to [Rg].

Proof. - By Lemma 4.4,

dimC(a) = [NG((T)/NGo((T)]~~~CO((T) = [WG(U)/WGO(~)][R~]

= [WG(~)/WG~(~)~[WG~(~)IW(~~)I= [W~(~)/w(%>l= [&I. n

THEOREM 4.16. - The operators {R(r, a), r E R,} form a basis for the algebra of intertwining operators of Ij2 (fl).

Proof - By Lemma 4.15 it suffices to show that the operators are linearly independent. Suppose that c,, w E R,, are constants so that

c G&W, df(4 = 0 WER,

for all f E XFI~(~),X E G. Fix w1 E R,. Then for all f = a($) with 4 E ?@(a) and all ~0 E Go, we have

c cwR(w, Q(xon,;) = 0. WER,

ANNALES SCIENTIFIQUES DE L’6COLE NORMALE SlJPl?RIEURE

126 D. GOLD3ERG AND R. HERB

Now by Lemma 4.13, R(w,a)f(zan;t) = 0 unless w = wrwa where w. E WQ(~) rl R, = Rz. Now again using Lemma 4.13,

Thus

c C wworlh, wo)R’(wo : 0)44x0) = 0 UJOERZ

for all 4 E ‘$,(a) and all zo E Go. Now since we know that the operators R”(wo : a) are linearly independent on 7-$(c), we can conclude that the c,,,~ ~(wr , WO) and hence the c wlwo are all zero. n

As in Arthur [2] we now have to deal with the cocycle q of Lemma 4.12. Fix a finite central extension

over which 7 splits. Also define the functions & : A, + C* and the character x0 of 2, as in Arthur [2]. Then we obtain a homomorphism

& 0) = J;‘(r)R(r, o-), T E R,,

of ii, into the group of unitary intertwining operators for .IP((T) which transforms by

ii(z?-, 0) = x&-1iz(r,o), .z E z,, r E R,.

Now we can define a representation R of R, x G on ‘HP(~) given by

Let n(fi,, xc) d enote the set of irreducible representations of fi, with 2, central character X~ and let IfI, (G) denote the set of irreducible constituents of Ip(o).

THEOREM 4.17. - There is a bijection p H r,, ofII(fii,, xc) onto III,(G) such that

R=$ fEW~o,Xo) (P” 8 %I.

Proof. - Write the decomposition of R into irreducibles as

4e S6RIE - TOME 30 - 1997 - No 1


where p runs over II(k,, x~), 7r runs over III,(G), and each rn,,* 2 0. This corresponds to a decomposition

where VP” and IV, denote the representation spaces for the irreducible representations p” and rr. For each p, we also write

xp N VP” @ c wp-

for the p-isotypic component of tip(a). Each subspace ‘HP is invariant under the action of R, in particular by all of the intertwining operators fi(r, g), T E fig. Since these intertwining operators span the space C(a) of G-intertwining operators for ‘HP(~), each T E C(g) must satisfy T(‘FI,) C Xp for every p.

We will first show that given n, there is at most one p such that mp,rr > 0. So fix x and suppose that there are p1 and pa such that rnPlrn > 0, i = 1,2. Then W, occurs as a G-summand of both I-& and ‘Flp~. Thus there is T12 # 0 in H~m~(‘Flpl, ‘Flpz). We can extend T12 to an element T of C(a) by setting T = Tl2 on tiPI and T = 0 on R,,, p # ~1. Thus there is T in C(o) such that T(‘l-l,,) c XpI,,. But by the remark in the previous paragraph, T(7ip,) c ‘H,,,. Now since T(X,,) # 0, we must have p1 cv p2.

Now fix p and look at tip 2~ V,V 18 W where W = C, W,“P,z. We will show that W is irreducible as a G-module. Thus suppose that W = WI $ W2 where WI, W2 are G-submodules of W. Then we can define T E C(a) by T(u @ (~11 + ~2)) = %r @ w1 if v E V,V, wt E WI, wa E W2, and T = 0 on tiff if p’ $ p. Now since T E C(a), we can write T = C,. c,%(r, P) where T runs over fig. Thus for all v E VP”, w1 E WI, w2 6 W2, we have

v @ w1 = T(v @ (WI + ~2)) = (c c,p”(+) @ 101 + (c c,p”(+) @ w2.

r T

Suppose W2 # 0. This implies that c, c,p”(r)v = 0 for all ‘u so that w @ WI = 0 for all wl. Thus WI = (0) and hence W is irreducible. But this implies that mp,?r 5 1 for all 7r and that there is at most one 7r such that mP,r = 1.

Define III = {p E II(il,,xc) : mp,x = 1 for some 7r).

For each p E It1 we have shown that the representation 7r such that mp,x = 1 is unique. Thus we will call it 7rp. Further, we have shown that 7rpl N rP2 just in case p1 11 p2. Further, by definition of l&,(G), each 7r E II,(G) occurs in ?fP(cr) and so must be of the form rP for some p E III. Thus to complete the proof of the theorem we need only show that II, = II(&,x,).

Since

ANNALES SCIENTIFIQUES DE L’kOLE NORMALE SUPfiRIELJRE


we must have dimC(a) = CPEnl (deg p)“. But since the R(T, a), T E R,, form a basis for C(a), we know that

dimC(a) = [R,] = c (deg P)“.

Thus II1 = II& Xg). n

REMARK 4.18. - Suppose now that G/Go is cyclic. Then in [ 11, Arthur predicts a dual group construction of a group RG,~, in terms of the conjectural parameter $ for the L-packet of LT, which should also describe the components of Ind$(a). In particular, R* ,o = w*,~Iw&7~ where these groups are defined in terms of centralizers of the image of $. Furthermore, Arthur has conjecturally identified Wz,C with W(@,) and W+,, with WC(g). Thus, if the conjectural parameterization exists in the connected case, and Shelstad’s Theorem [ 121 extends to the p-adic case, then it must be the case that R, N R+,,, . That is, we have shown that there is a group side construction of Arthur’s R-group, if such an object exists. (For more details see [l] and [6], particularly Sections 1 and 4.)

5. R-groups for Indg (a)

In this section we will study representations of G which are induced from discrete series representations of a parabolic subgroup P of G. Thus we revert to the notation that parabolic subgroups of Go are denoted by PO. Let P = MN be a cuspidal parabolic subgroup of G.

Let c be an irreducible discrete series representation of M and let ~0 be an irreducible constituent of the restriction of g to MO. We want to find a basis for the intertwining algebra C(a) of the induced representation bids(a). Since 0 is contained in Ind$ ((TO) we know Indz(o-) is contained in Indg (Ind$ (go)) N Ind$ (GO). In 54 we found a basis for the intertwining algebra C((T~) of Ind$(co). We will see how to obtain a basis for C(a) by restricting the intertwining operators defined in $4.

We first need to embed cr in a family cr,, v E a&, where a is the real Lie algebra of the split component A of M. Write X(M), X(A) for the groups of rational characters of M, A respectively. Let

r-:X(M)@zR+X(A)@zR

be the map given by restriction. That is r-(x @ t) = XIA @ t for x E X(M), t E R.

LEMMA 5.1. - The homomorphism T : X(M) @Z R + X(A) 63~ R is surjectiue.

Proof. - Since G is a linear group we have an embedding of G in L = GL(V), where V is a finite dimensional F-vector space. Since A is a split torus, the action of A on V can be diagonalized. For any x E X(A) let V(x) = {U E V : av = x(u)” for all a E A}. Let xi, 1 < i 5 k, denote the distinct elements of X(A) such that V, = V(xi) # (0). We can identify a E A with the block diagonal matrix with diagonal entries Xi(u)ld, where di = dim K and Id, denotes the identity matrix of size d,, 1 5 i < k.

4e S!?RtE - TOME 30 - 1997 - No 1


Since M = CG(A), we have M c CL(A) N GL(Vl) x GL(V2) x . . . x GL(Vk). For each 1 5 i 5 Ic we can define deti E X(M) by deti(mr, . . ..mk) = det mi. NOW det; @d;’ E X(M) @Z R and for a E A, deti @d;‘(a) = x~(u)~” @ dil = xi(a) 8 1. Thus r(deti 8&l) = xi 8 1. The xz, 1 < i 5 k, are generators of X(A), although they need not be independent. Thus T is surjective. n

Let X0(M) = {x E X(M) : X]MO = 1).

LEMMA 5.2. - The kernel of T is X0(M) @Z R.

Proof. - Suppose X @ t is in the kernel of T where X E X(M) and t E R. If t = 0 then x @ t is the identity element. Assume t # 0. Then Ix(a)]> = 1 for all a E A implies that ]~(a)]~ = 1 for all a E A. Since X[A is a rational character of a split torus this implies that xIA = 1. But restriction from X(MO) to X(A) is injective [13, Lemma 0.4.11, so that Xln;ra = 1. Thus X E X0(M). n

Recall the homomorphism HATO : M” --+ Hom(X(M’), Z) defined by

< Hw(m),x >= log,Ix(m)lF, m E MO, x E X(MO).

Define an analogous homomorphism Hhf : M --f Hom(X(M), Z) by

< HM(m), x >= h,lx(m)lF,m E M, x E X(M).

LEMMA 5.3. - Suppose that x E X0(M). Then < HM(m), x >= 0 for all m E M.

Proof. -Let X E X0(M). Thus x(mo) = 1 for all mo E MO. Let d be the index of M” in M. Thus md E M” for all m E M so that x(md) = 1 for all m E M. Thus x(m) is a dth root of unity and Ix(m)lF = 1 for all m E M. Thus < HM(m),x >= Zog,Jx(m)lF = 0 for all m E M. m

Recall that Hom(X(M’), Z) @Z R N Hom(X(A), Z) @Z R = a is the real Lie algebra of A, a* = X(A) I& R is its real dual, and a* --c = a* @n C is its complex dual. For each v E a& we have a character X”, of MO defined by X”,(m) = Q<~.@(~)+‘>, m E MO. By Lemmas 5.1 and 5.2 the mapping r above induces an isomomorphism

X(M) @z c ‘* : x0(M) @z c

11 X(A) @z c c” a&.

By Lemma 5.3, for each m E M, H Thus for each v E a;,

M m is an element of the complex dual of $O\~~~~~. ( ) we can define a character xy of M by

x,(m) = q<fh(+;‘(“)>, m E M,

LEMMA 5.4. - For all u E c& the restriction of xv to M” is x”,.

Proof. - For mo E MO, x E X(M) we have

< HM(mo), x >= b,Ix(mo)lF = b,lxdmo)lF =< HMO(w), x0 >

where Xc denotes the restriction of x to M ‘. Since the isomorphism T* comes from the restriction map it is easy to see that < HM(mo),r;‘(v) >=< HMO(mo),v > for all m. E ni”,u E c&. n

ANNALES SCIENTIF’IQUES DE L’hCOLE NORMALE SUPBRIEURE


As above, let CJ be an irreducible discrete series representation of M and let ~0 be an irreducible constituent of the restriction of Q to MO. Let Vo be the representation space for 00 and let

W = {f : M + V. : f(mmo) = c~~(rno)-‘f(m) for all m E M,VLO E MO}.

Then M acts on W by left translation and we will call this induced representation (IM, IV). Let V denote the representation space for CT and fix a non-zero intertwining operator S : V + W so that So = 1fil(rn)S for all m E M. Since I, is unitary, we can also define a projection operator P : W -+ V so that P1~(m) = g(m)P for all m E M and PSv = v for all v E V. We also define representation spaces

7-&(a) = {c$ E C”(G, V) : $(zmn)

= n’F4(m),(m)-1$(x) for all 2 E G, m E M, n E N};

‘Flpo(ao) = {$ E C”(G, VO) : $(zmon)

= 6,0i(mo)ao(mo)-1$(s) for all x E G, mo E MO, n E N};

Xp(1~) = {$ E C”(G, W) : $(zmn) 1

= 6,’ (m)l~~(m)-l$(x) for all II: E G, m E M, n E IV}.

In each case G acts on the representation space by left translations and we call the induced representations Ip (0)) 1~0 (a~), and IP(lM) respectively. They are the representations Ind$(a), Ind$ (go), and Indz ( IndzO (go)) respectively.

The intertwining operators S : V + W and P : W --+ V induce intertwining operators S* from (Ip(a),Xp(a)) to (IP(~M),~-IP(IM)) and P* from (IP(~M),~~P(~M)) to (IP(~), tip(a)) given by

(S*q4)(z) = S$(z) for all $J E ‘Hp(c~),z E G;

(P*$)(x) = Pqb(x) for all II, E Tip(l~),z E G.

There is also an equivalence T between (Jp(1~), ‘Flp(l~)) and (1~0 (CO), ‘Flpo (a~)) given by

(T$)(rc) = $(x)(l) for all $J E Xp(l~),z E G.

Its inverse is given by

(T-‘@‘)(z)(m) = S$(m)$(zm) for all q!~’ E 7ip(c~o),~c E G, m E M,

Recall for each v E C& we have defined characters x”, of M” and xv of M such that x”, is the restriction of xv to MO. We use these characters to define representations a(v) = a@xy and 1~ (v) = IM EC xv of M and Ok = o. @ x0(v) of MO. As above we use these to form induced representation spaces Ep(cr, V) = X~lp(a(v)), ‘HP(IM, V) = XP(~M(V)), and ‘HP0 (a, V) = EFlpo (crO( v)). The intertwining operators S : V + W and P : W --+ V also intertwine O(V) and Ihl (v) and so as above define induced intertwining operators

4e SBRIE - TOME 30 - 1997 - No 1


s: : xp(o,v) -+ XP(IM,V) and PC : Xp(IM,v) + T-f~(a,v). There are also equivalences T, : x~(I~,v) ---t XPO(CJO,V) given by

(T,@)(z) = $(x)(l) for all 1c, E ‘HP(~M, v),z E G.

The inverses are given by

(T;‘$‘)(d(m) = &m)xv(m)ll’(4 f or all +’ E Xpo(go, V),X E G, m E M.

Suppose PI = MN1 and P2 = MN2 are two cuspidal parabolic subgroups of G with Levi component M. In $4 we defined a meromorphic family of intertwining operators

J(P,o : P,” : 00 : V) : 3-lp+Jo,~) ---t ‘FlP+o?).

We can transfer these intertwining operators to the equivalent spaces 7fp, (lM, v), i = 1,2, by means of the equivalences TQ,. Thus we define

J( P2 : PI : IM : V) = TV$ J( P; : P,” : o. : v)T,,~~.

We can also define

J(Pz : PI : o : V) : 3-lp, (a, V) --) xp, (a, V)

J(Pz : PI : CJ : V) = Pv*,p*J(P2 : PI : IM : 2+9;,,.

LEMMA 5.5. - Suppose that v E &(P~ : P,“) so that J(Pi : PF : ~0 : V) is given by the convergent integral

J(P,o : Pf : 00 : V)?f(Z) = J

rv nN $+?t)dqx E G,# E 7fp+o,v).

1 2

Then J(P2 : PI : IM : V) is given by the convergent integral

and J(P2 : PI : CJ : V) is also given by the convergent integral

ANNALES SCIENTIF’IQUES DE L’kOLE NORMALE SUPfiRlEURE


Proof. - Using the definitions of the operators and the transformation property of the representation space X~lp, (I M, v), we have for all z E G, m E kf, $ E ~-IP, (IM, v),

J(Pz : Pl : l&f : Y)?j(X)(rn) = T&qP,o : P; : (To : v)T,,p,~(z)(m)

= Xv(f+&(m)(J(P. : p1” : 00 : q/,P,$)(xm)

Now there is a homomorphism p : M 3 R+ so that for all m E M, f E Ccm(~l n Nz),

J ’ N nN, f(mEzm-l)dE = /3(m)

1 2 J Iv “N‘ f@P. 1 2

For m E M” we know that

,/3(m) = Sk1 (m)6F2i (m).

Since ,86~I~6& is a ho momorphism from the finite group M/M0 into R+, it must be identically one. Hence for all m E M we have

Thus J(P, : PI : 1~ : v)+(z)(m) = .I, nN $(zc)(m)fi.

1 2

Now for x E G, m E M, q5 E ‘Hp,(a, v), we have

J(P, : PI : cJ : v)qs(x) = [P;,&J(Pz : PI : I, : v)s;,pl~](x)

= P [J(P, : PI : IM : z+~,pl~](x)

=p. . .!

‘iiJ nN‘ K,P141 @)a 1 2

=PS. J N "N, dx3fi 1 2

Let 7-(P,” : P,O : CTO : V) be the scalar normalizing factors used in 54 to define the normalized intertwining operators

R(P,O :P; :oo) = .(P," :P," :cro :0)-kJ(P,o :P; :CJ~ :O).

4’ Si?.RlE - TOME 30 - 1997 - No 1


The fact that

,(P,” : Pf : 00 : zq’J(P,” : Pp : 0‘0 : v)

is holomorphic and non-zero at v = 0 and that Ty,pt, i = 1,2, are equivalences, will imply that

,(P,” : Pf : 00 : v)-’ J(P, : PI : I&f : v)

is also holomorphic and non-zero at v = 0. Thus we can define

R(Pz : PI : IM) = ,(P,” : Pf : 00 : 0)-‘J(P, : PI : IM : 0).

We also define

R(Pz : PI : cr) = T(P; : Pp : fJ’0 : 0)-‘J(Pz : PI : c7 : 0) = P*R(P, : PI : I&f)s*.

LEMMA 5.6. - Let q5 E Tip,(a). Then R(P2 : PI : IAJ)S*~(Z) E S(V)fir all x E G.

Proof. - For every v we have an intertwining operator J(P2 : P1 : IM : v)Sc : 3-IP, (CT, u) + 7-b* (Ill4 : v). In order to carry out arguments using the integral formula and meromorphic extension of the intertwining operator we want a compact realization of the representation. Since we do not know if there is a maximal compact subgroup of G which meets every connected component, we proceed one coset at a time. Let GM = GoM and write G = U~=,xiG~~. Then P C Ghf for any parabolic subgroup P of G with Levi component M. Let K” be a good maximal compact subgroup of Go so that Go = K”Po. Thus Gnl = K’P. Let

7&+) = {fK E C”(KO, V) : f$y(kmn) = Cl(rn)fK(k) forallmEK”~~,nEKo~N,IcEKo}.

For any $ E ‘Flp(a : u) and 1 2 i < k we can define

$(x), if z E xiG,+l; h(x) = { o otherwise.

Then $i E ‘7f~lp(a : .v) for each i and C#J = ~~=, &. Thus every element of ‘HP(~ : V) is a sum of elements supported on a single coset of G M in G and so it is enough to prove the lemma for C$ E ‘Flp, (CJ) supported on a single coset of GM in G.

Fix 1 5 i < Ic and define

F;(U) : IFlp,(C 1 U) + ti~O(CT)

by Pi(v)+(k) = +(Xi/?) for all k E K”. Define

Iy(u) : lijyo(iT) -+ 3-l&7 : u)

by

K1W.fd4

= s,~(rn)a-‘(m)X,‘(m)~~(k), if x = xikmn,k E K”,m E M,n E N; 0, otherwise.

ANNALES SCIENTIFIQUES DE L’BCOLE NORMALE SUPliRIEURE


Then Fi(v)FiP1(v)fK = fK f or all .f~ E ‘FIKo(~) and J?-‘(Y)F~(v)~ = $i for all (b E ?&,(u : v).

Fix 1 L i < k and 4 E EP, (u) = ~-IP, (g : 0) such that $ = $i is supported on x~GM. Let .f~ = Pi(O)4 E XKO(CT) and for each v define &(u) E 3-IP,(g : V) by &(v : x) = Fi-l(v)f~(x),x E G. We have 4;(O) = FiP’(0)Fi(O)$ = & = $.

Fix w* E S(V)l and define @(Y : z) =< J(P2 : PI : IM : v)S,*&(v : x), w* >. Then v ++ @(Y : x) is a meromorphic function of Y E a& for each x E G. If v E c&P; : P,“), by Lemma 5.5 we have

J(P2 : Pl : Ifif : I@;#+ : x) = s

S&(Y : xE)dFi E S(V). N1 nN2

Thus Q(x : V) = 0 for all v E &(Pi : Pj’) and hence for all V. Thus

J(Pz : PI : IM : +‘;c$~(, : x) E S(V)

for all v and so

,(P,” : Pl” : go : zl)-’ J(P, : PI : IIM : w)S:&(~ : x) E S(V)

for all v E L&,X E G. In particular for v = 0 we have R(P2 : Pl : Ihl)S*$(x) E S(V) for all x E G. n

COROLLARY 5.7. - Let 4 E tip,(a). Then R(P, : Pl : IM)S*C$ = S*R(P2 : Pl : o)&

Proof. - This follows from Lemma 5.6 since SP is the identity on S(V). n

LEMMA 5.8. - Suppose PI, Pz, and P3 are cuspidal parabolic subgroups of G with Levi component M. Then

R(Pl:P3:I~)=R(Pl:P2:Ihl)R(P2:P3:IIM)

and R(Pl : P3 : a) = R(Pl : Pz : g)R(P2 : P3 : o).

Proof. - The statement for IM follows easily from Lemma 4.8 since the intertwining operators T, are equivalences. Now using Corollary 5.7 we have

R(Pl : P3 : cr) = P*R(Pl : P3 : IIM)S* = P*R(Pl : P2 : IM)R(P2 : P3 : IM)S*

P*R(Pl : Pz : IIM)S*R(P2 : P3 : a) = R(P, : Pz : g)R(Pz : P3 : g). n

LEMMA 5.9. - Suppose PI and P2 are parabolic subgroups of G with Levi component M. Then R(P, : PI : a) is an equivalence from tip1 (o) onto Yl-l~lp, (a).

Proof. - The proof is the same as that of Lemma 4.9. H Now as above we define NG(~~) = {g E.NG(A) : ui Y 00). If w E W~(cra) =

N&o)/M”~ go can be extended to a representation ga,w of the group Mz generated by MO and any representative n, for w. Define T(n,) : W -+ W by

T(nw)f(m) = ao,w nw ( )f(~,l~~tJ,~ E M.

4e SkRIE - TOME 30 - 1997 - No 1


It is easy to check that

T(n,)l~(m) = l&w-rm,l)T(n,), m E hf.

Next we define intertwining operators BP(W) : K,-~P~(~M) + XFIP(~IM) by

&+)$J(g) = %L-lPw %J ( $%Mgnw),g E G.

Finally we define self-intertwining operators R(w, 1~) : XP(IM) -+ EP(~M) by

R(W,IM) = Bp(wJ)R(W-lPW : P : IM).

Recall that in $4 we defined intertwining operators

AP(W) :Xw-m&o) + XP+O) and qw, 00) : %+o) -+ 3-IP(OO).

It is easy to check that

BP(w) = 5!‘;‘Ap(w)T,-1~, and R(w, I& = T$(w, ao)Tp

where Tp : ‘Flp(1~) -+ ‘Flpo(ao) is the equivalence defined above. In particular this implies that BP(W) and R( , I ) w M are independent of the coset representatives chosen.

Define No = {g E NG(M) : CJ~ N 0). If w E WG(~) = NG(u)/M, 0 can be extended to a representation of the group A&, generated by A4 and any representative n, for w. Denote such an extension by gu, and define A&(w) : 7-lFlw-~pw(o) + tip(o) by

LEMMA 5.10. - The intertwining operator Ah(w) is independent of the choice of coset representative n, for w E WG(cr). For ~1, w2 E WG(O) there is a non-zero constant cp(wl, w2) so that

Proof. - The proof is exactly the same as that of Lemma 4.11. Finally, for w E WG((T), we define R(w,c) : XF~~((T) + tip(o) by

R(w,a) = A&(w)R(w-lPw : P : c).

Note that for u E WG(oo) we could also have defined an intertwining operator

R’(u, 0) : tip(o) + T-&(a)

R’(u, 0) = P*R(u, IM)s*.

We want to relate these two definitions. Let

wG(oo, 0) = [N&o) f-l NG(O)]/M’ c WG(aO)-

n

ANNALES SCIENTIFIQUES DE L&COLE NORMALE SUP6RIEURE


Suppose that z E NG (aa) fl NG (F). Then z represents an element z&f0 E W~(cra, a) and an element XM E WG(U). Let

p : WG(~O,~) + wG(“)

be given by p(xM’) = zM,x E NG(~o) fl NG(~).

LEMMA 5.11. - The mapping p is surjective. Its kernel is WM(CJO) = NM (ao)/M”.

Proof. - Let w E lV~(g) and let x E NG (a) be a representative for w. Then a$ is contained in the restriction of g” 51 u to iVf” so that there is m E A4 such that (T; E (TF. Hence w has a representative xm-l E NG(c) n N~(cr~). Thus p is surjective. Clearly p(xM”) = M just in case x E NG(~o) fI NG(~) fl M = Nhl(0a). n

LEMMA 5.12. - Suppose that u E W, (co) is in the complement of WC ( 00, a). Then

R’(u,a) = 0.

of u E WG(oo, u), then there is a complex constant c so that

R’(u, a) = cR(p(u), ff).

Proof. - Using Corollary 5.7, for any u E WG(CJO),

R’(u, u) = P*R(u, I&f)s* = P*Bp(u)R(u-lPu : P : IM)s* = P*Bp(u)S*R(u-lPw : P : fl).

But for any g E G, 4 E XFI,- 1 Pll(a), if IC E NO is a representative for u,

P*&(u)s*&7) = ~u-lPu(x)3(PT(x)S)~(gx).

Since T(x) intertwines 1~ and I&, we see that PT(z)S intertwines c and g.” 21 cu. Thus PT(x)S = 0 and hence R’(u,c) = 0 unless g.” N g.

Suppose u E W~(g~,a.). Write u = u,, w = W, = p(u),z E NG(go) n NG(a). Then PT(z)S and gw (x) both intertwine g and gz, and gzu (z) # 0. Thus there is a complex constant c’ so that PT(z)S = c’cr,(x). Thus for any g E G,

R’h G47) = (Yu-lPu(x)~(PT(x)S)[R(?h-lPU : P : f+b](gz)

-

= c’(Y,-lpu x)” uU,(x)[R(U-lPU : P : a)qq(gz) I I = c A P (,)A( w-lPw : P : a)+(g) = c’R(w, ~>al>. n

LEMMA 5.13. - The R(w, cr), w E WG(~), span the algebra C(g) of self-intertwining operators on E~lp(ff).

Proof. - Let R be a self-intertwining operator for tip(c). Then S* RP* is a self- intertwining operator for XFIp(IM), hence in the span of the R(u,IM),u E WG(~O). But then R = P*S*RP*S* is in the span of the P*R(u,lht)S* = R’(u, g),u E WG((T~).

4e ShRlE - TOME 30 - 1997 - No 1


But by Lemma 5.12, each R’(u, a) is either zero or a multiple of one of the operators R(w, g),w E wG(g). w

LEMMA 5.14. - Let u E WG((T~) and suppose that R(u,l~) is scalar. Then u E W&&a) and R(p(u),o) is scalar.

Proof. - Suppose that there is a constant s E C such R(u, IM)$ = s$ for all $ E xFIp(1~). Since R(v, IM) # 0, s # 0. Now for all 4 E %~(a),

R’(u, o)$ = P*R(u, IM)s*$ = sp*s*4 = sfp.

Thus R’(u,~) is scalar and non-zero. Thus by Lemma 5.12 we have u E WG(aa,fl). Further, by Lemma 5.12, there is a constant c so that R’(u,~) = &@(~),a). Since R’(u,~) # 0, c # 0. Thus R(p(u),a) = C-lR’(~,a) is scalar. w

LEMMA 5.15. - There is a cocycle q so that

for all Wl,W2 E wG(G).

Proof. - The proof is similar to Lemma 4.12. w As in $4, if W,!$ (go) is the subgroup of elements u E WGo (00) such that R”(u, ~0) is

scalar, then W$(ao) = W($) is generated by reflections in a set +r of reduced roots of (G, A). Let cP+, @t be defined as in $4. Since M centralizes A, WG(O) c NG(A)/M acts on roots of A and we can define

R, = {w E WC(~) : w/l E Q’ for all ,D E at}.

We want to prove the following.

TEOREM 5.16. - The R(w, a), w E R,, form a basis for the algebra of intertwining operators of Ip(g).

In order to prove Theorem 5.16, we will first compute the dimension of C(a) using our knowledge of the dimension of C(U,). We denote the equivalence classes of Indg (a) and Ind$,(co) by iG,M(Cr) and iG,IMO(‘Ta) respectively. Let X and Y denote the groups of unitary characters of G/Go and M/M0 respectively. For x E X, let XM E Y denote the restriction of x to M. Define

X(4 = {x E x : xhf @ 0 N a}; xl(a)= {x c x :X@bG,M(+ '1G,M(fl)};

Y(0) = {T/ E Y : 0 @ 7 N 0).

Let s denote the multiplicity of go in the restriction of (T to MO.

LEMMA 5.17

dim C(ao) = dim C(a)s2[X/X(g)][X1(g)/X(~)].

ANNALES SCIENTIF'IQUES DEL'~~COLENORMALESUP&IEURE


Proof. - Using Lemma 2.13,

iM,M+O) = s c 0 63 q. MY/Y(v)

This implies that

Since both G/Go and M/M0 are finite abelian, it is clear that the map X H XM induces an isomorphism between X/X(a) and Y/Y(c). Thus we can rewrite

iG,M(iM,M+O)) = s c iG,M(a @ XM).

XEXIX(C)

But by Corollary 3.3 the induced representations ic,M(a @ XM) are either disjoint or equal. Further,

iG,M((T ‘8 XM) = iG,M(a) @ x = iG,M(c)

just in case X E X1 (0). Thus we have

iG,M(iM,M+O)) = s[xl(o)/xb)] c iG,M(d @ x

XEXlX1 (g)

where the representations ‘~G,M (a) $9 X are disjoint for X E X/Xl (0). Thus

dimC(aa) = dimC(a)s2[Xl(a)/X(o)12[X/X1(g)] = dimC(a)s2[X/X(a)][X1(c)/X(g)].

LEMMA 5.18

s”[~/~(~>l[~~(~)l~(~)l = [WG(~O)IIWG(~)I. Proof. - First, using Lemma 2.13 we have

S’[x/x(C)] = S”[Y/+)] = [NM(gO)/M”] = [wM(gO)]*

We claim that

This would establish the lemma since by Lemma 5.11 we have

n

[wG(fl)] = [wG(crO> fl)/wM(cTO)I.

4e SfiRlE -TOME 30 - 1997 - No 1


We will define a bijection between X1 (a)/X(a) and

Let X E X1(g). The equivalence class of r~ @ Xh{ depends only on the coset X of X in x,(c~)/X(a). Further, by definition of X,(g), we have

iG,M(fl@ XM) = iG,M(u).

By Corollary 3.2 there is z E NG(A) such that (T @ XM II c2. Thus ~.(Mo II O”]MO. Thus a; occurs in (T] MO and so there is m E M such that at 31 ar . Then y = zm-l E NG (CO). Although y E NG (a~) is not uniquely determined by X,

u @ XM z ffy1 N uy2

if and only if yly;’ E NG(gO) nlvG(g). Thus for each X E Xr(a)/X(a) there is a unique coset z(x) = (NG(uo) n N G u ( )) 5 in (NG(uo) nivG(0))\NG(Uo) such that XM @fl % U”. Finally, given z E NG(DO), a” is a constituent of ~M,MO(O,“) 2~ ~M,MO((T~) SO that there is 11 E Y such that (T” % (T (8 77. Now let X E X such that XM = n. Then g @ XM N a” so that ;Z; = Z(X). n

Recall from $4 that wG(go) is the semidirect product of subgroups R,, and w(@~) where R(w, go) is scalar for w E W(+I) and the R(r, (TO), T E R,,, give a basis for C(aa).

LEMMA 5.19

dim C(c) = [~~(~)l/[W(%)l.

Proof. - Combining Lemmas 5.17 and 5.18 we have

dim C(aa) = dim C(a) . ‘FG((!!A].

But from Lemma 4.15, dimC(aa) = [I&] = [WG(O~)]/[W(Q,)]. I can be naturally embedded in WG (A) = iv,( A)/MY Since W(@l) c NGo(A)/MO, ‘t

LEMMA 5.20. - WG(~) is the semidirect product of W(@,) and R,. For w E WG(C), R(w, 0) is scalar if and only if w E W((a,).

Proof. - If 5 E NGO (go) represents an element of w((a,), then by Lemma 4.14 R(‘LL,,IM) is scalar. Thus by Lemma 5.14, w, E wG((T) and R(w,,o) is Scalar. Let IV:(a) denote the set of all w E WG(~) such that R(w, O) is scalar. By the above W(aI) c W,$(c). Using Lemmas 5.13 and 5.19 we see that

[w~(~)lI[W(%)l = dim C(o) 5 W~(~)l/[%i!(~)l.

Thus W((a,) = w;(a).

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Now as in the proof Lemma 4.14, W(@r) is a normal subgroup of IV,(g) and so u&r = @r for all w E WG(~). This implies that

which yields the semidirect product decomposition. n

Proof of Theorem 5.16. - It follows from Lemmas 5.13, 5.15, and 5.20 that the R(w! a), 20 E R,, span the algebra C(a). Further, by Lemmas 5.19 and 5.20,

dimC(o) = [WG(O)]/W(@I)] = [&I.

Let r] be the cocycle of Lemma 5.15. Exactly as in $4 we can fix a finite central extension

over which v splits, a character x0 of Z,, and a representation R of fii, x G on tip (a). Let II(k,, xc) denote the set of irreducible representations of k, with 2, central character xc, and let II,(G) denote the set of irreducible constituents of IP((T).

THEOREM 5.21. - There is a bzjection p I-+ 7rp of II(&, xc) onto III,(G) such that

Proof. - The proof is exactly the same as that of Theorem 4.17. n

6. Examples

For applications involving comparisons of representations between groups and twisted trace formulas it is customary to use the following definition of parabolic subgroup. Let P” be a parabolic subgroup of Go. Then P = NG (PO) is a parabolic subgroup of G. Thus, using this definition, parabolic subgroups of G are in one to one correspondence with parabolic subgroups of Go. We will show that the parabolic subgroups obtained using this definition are also parabolic subgroups using the definition of $2. However they are not cuspidal in general. Indeed, recall from Proposition 2.10 that if PO is a parabolic subgroup of G, then the corresponding cuspidal parabolic subgroup is the smallest parabolic subgroup of G lying over P ‘. On the other hand, if P is any subgroup of G with P fl Go = PO, then P c NG( P”). Thus NG (PO) will be the largest parabolic subgroup of G lying over PO. We will give examples to show that this class of parabolic subgroups, which we call N-parabolic subgroups (N for normalizer), do not yield a nice theory of parabolically induced representations of G.

LEMMA 6.1. - Let P” be a parabolic subgroup of Go. Then P = NG(P’) is a parabolic subgroup of G. It is the largest parabolic subgroup lying over PO. If MO is a Levi component for PO, then M = NG(M’) n P is a Levi component for P.

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Proof. - Let PO = M”N be a Levi decomposition of P” and define M = NG(M’) n P. Then M n Go = M” and MN c P. Let z E P = NG(P’). Then zM”x-’ is a Levi subgroup of P” and so there is n E N such that zM”z-’ = nM”npl. Now TL-~IG E NG(Mo) n NG(Po) = M and SO z E NM = MN. Thus P = MN.

Let A be the split component of M ‘. Then M normalizes A and we define a Weyl group W = M/CM (A) w h ere CM(A) denotes the centralizer of A in M. Since M” c Cnf(A), we know that W is a finite abelian group. The split component of M is A’ = {a E A : m~-~ = a for all z E M} = {u E A : wa = a for all a E IV}. Let M’ = C, (A’). If we can show that A’ is the split component of M’, then A’ is a special vector subgroup.

Let a+ = @(PO, A) denote the set of roots of A in PO, a the real Lie algebra of A, and a+ the positive chamber of a with respect to a+. Fix w E W and define a, ={HEa: wH = H}. Since M normalizes PO, we have w++ = a+ and wa+ = a+. Let k be the order of w. Then since a+ is convex, for any H E a+ we have H~=H+~H+~~H+...+w~-~H~~+withwEi,=H,.Thus~~=~,n~~#0 and so for any o E a+, the restriction of Q: to aul is non-zero. Since W is a finite abelian group, an easy induction argument shows in fact that the restriction of o to a’, the real Lie algebra of A’, is non-zero for every cr E @J +. Thus M’ n Go = CGo (A’) = CGO (A) = MO.

Let A” be the split component of M’. Thus A’ c A”. But since M’ n Go = MO and M c M’, we have A” = {u E A : zaz-’ = a for all z E M’} c A’ . Thus A’ = A” is the split component of M’. This implies that A’ is a special vector subgroup and that M’ = CG(A’) is a Levi subgroup of G. But since the restriction of LY to a’, is non-zero for every Q! E a+, we can choose a set of positive roots (a’)+ of L(G) with respect to L(A’) by restricting the roots in a+. With this choice of positive roots, we obtain a parabolic subgroup P’ = M’N’ of G with N’ = N. Thus M’ normalizes N. It also normalizes M” since M’ n Go = MO. Thus M’ c NG( MO) n NG (PO) = M. Now M’ = M SO

P’ = MN = P. n Let (PO”, Ao) be a minimal p-pair in Go and let A denote the set of simple

roots of A0 in Pt. Then as usual the standard parabolic subgroups of Go are indexed by subsets 0 of A. Write (Pg, A@) for the standard parabolic pair of Go corresponding to 0 C A and write PO = N,(P$), the N-parabolic subgroup of G lying over Pg. Let NG( Pi, Ao) be the set of elements in G that normalize both A0 and Pt. Clearly NG (P{, A,) n Go = Pt n NGO (Ao) = CQ (Ao) = Moo. Write WG(P~,AO) = NG(P~,Ao)/M~. Then WG(P~,AO) acts on A and for each 0 c A we write

W(0) = {w E Wc(P&Ao) : w0 = 0).

LEMMA 6.2. - For all 0 C A,

PO = U,,W@)WP&

Proof. - This follows from Lemma 3.8. n Now that we have a simple method of computing the groups PO, we will give examples

to show the following unpleasant facts.

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FACT 6.3. - Let PF and P$ be parabolic subgroups of Go and let Pi = NG(P,O), 1; = 1,2. Then Pf c Pt does not imply that PI c P2.

EXAMPLE 6.4. - Let G = 0(2n), n 2 2. Then Go = SO(2n) and the minimal parabolic subgroup B” of Go is the group of upper triangular matrices in SO(2n) with A0 the subgroup of diagonal matrices. The simple roots are

A = {el - es, e2 - es, . . . . en-l - en, e,-l + e,}.

The Weyl group WG(B’, Ao) has order two and is generated by the sign change G2 : e, H -e, that interchanges e,-r - e, and e,-l + e,. Thus a subset 0 of A is stable under c, just in case neither or both of e,-1 f e, belong to 0. So for example if 0 = {e,-r - e,}, then B” C Pi, but B = B” U c,B” is not contained in PO = Pg.

We call PO a minimal N-parabolic subgroup of G if given any N-parabolic subgroup P of G there is LC E G such that PO c xPx-l.

FACT 6.5. - G need not have a minimal N-parabolic subgroup. Suppose PO is a minimal N-parabolic subgroup of G. Then it is easy to see that

P,O = PO n Go is a minimal parabolic subgroup of Go. Now by Lemma 6.2, PO meets every connected component of G. But as Example 6.4 shows, there are N-parabolic subgroups of O(2n) which are contained in the identity component SO(2n). Thus no conjugate could contain the minimal N-parabolic subgroup.

If P” = MON is a Levi decomposition for P ‘, then we obtained a Levi decomposition P = MN for P = NG( P”) by defining M = NG(M’) n P. Thus A4 depends on both &lo and P.

FACT 6.6. - Suppose that PI = MrNr and P2 = M2N2 are N-parabolic subgroups of G such that Mp = M:, i.e. PF and P$ have the same Levi subgroup. Then it need not be true that n/r, = A42, or even that Mr and iVl2 have the same number of connected components.

EXAMPLE 6.7. - Let G = O(8) as in Example 6.4. Let PI = PO, where Or = {es - ed} and let Pi = PO, where @a = {er - ea}. Then PI = Pp is connected and P; = (P;)“uc~(P~)o meets both components of G. Let w = (13)(24) E NG(Ao)/Ao be the Weyl group element that permutes the pairs (er, es) and (ea, e4) and define PZ = wP;w-‘. Then wAOzu~-l = A@, and so Pt = w(P~)~Iu-’ and Pf both have Levi component M; = CGo(Ao,). H owever iklr = A@ is connected and M2 = i@’ U c2Mi meets both components of G.

In addition to structural problems, the class of N-parabolic subgroups does not yield a nice theory of parabolic induction. One of the basic cornerstones of representation theory in the connected case is that every irreducible admissible representation is contained in a representation which is parabolically induced from a supercuspidal representation and every tempered representation is a subrepresentation of a representation which is parabolically induced from a discrete series representation. But if supercuspidal and discrete series representations are defined as in the connected case and in $2, then the Levi component M of a parabolic subgroup P has no supercuspidal or discrete series representations unless P is cuspidal. Thus we will not in general be able to obtain all irreducible admissible or tempered representations of G via induction from supercuspidal or discrete series representations of N-parabolic subgroups.

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EXAMPLE 6.8. - Define G = O(2) = SO(2) U wSO(2) as in Remark 2.2. The only parabolic subgroup of Go = SO(2) N FX is itself. Thus the only N-parabolic subgroup of G is Nc(Go) = G itself which has split component Z = { 1). But G has no representations with compactly supported matrix coefficients, hence no supercuspidal representations. It also has no representations with square-integrable matrix coefficients, hence no discrete series representations.

As can be seen in Example 6.8, the problem with using the standard definitions for supercuspidal and discrete series representations with N-parabolic subgroups P = MN is that the split component of M may be smaller than the split component of 44’. In order to guarantee the existence of enough supercuspidal and discrete series representations we could define a representation of M to be supercuspidal (respectively discrete series) just in case its restriction to MO is supercuspidal (respectively discrete series). Then it would be easy to prove as in Theorem 2.18 that every irreducible admissible (respectively tempered) representation of G is contained in a representation which is induced from a supercuspidal (respectively discrete series) representation of an N-parabolic subgroup.

Another basic property of parabolic induction in the connected case is the following. Suppose that PI = Ml Nr and P2 = M2N2 are parabolic subgroups and rsi, i = 1,2, are irreducible representations of Mi which are both either supercuspidal or discrete series. Then if the induced representations Indg% (gi) are not disjoint, then the pairs (Ml; ol) and (Mz, ~72) are conjugate. Further, in the discrete series case, the induced representations are equivalent. These properties fail in the disconnected case when the P; are N-parabolic subgroups of G and supercuspidal and discrete series representations are defined as above.

EXAMPLE 6.9. - Let G = O(8) and define PI and P2 as in Example 6.7. Recall in this casethat Ml = M,O N GL(~)xGL(~)~ while M2 = M~uc~M~ N GL(2)xGL(l)x0(2) with M,!j = GL(2) x GL(1) x SO(2) = MF. L e (~0 = p @ x1 ~3x2 be an irreducible unitary t supercuspidal representation of MF = Mt where p is an irreducible unitary supercuspidal representation of GL(2), x1 is a unitary character of GL( l), and x2 is a non-trivial unitary character of GL( 1) with x; = 1. Then (~0”” = p @ x1 @ xT1 = (T~ so there is an irreducible representation (~2 of n/r, which extends go. Further,

Ind$ (UO) = 02 B (~2 8 11)

where 7 is the non-trivial character of Mz/M$‘. The representations c2 and g2 @ n of M2 would both be supercuspidal (and discrete series) since they both restrict to p. on M:. Now

Indg;(oa) N Ind~~Ind~~(rra) = Indgz(cr2) $Indgz(a2 87).

Since PF and P; are parabolic subgroups of Go with the same Levi component M: = Mt we have

Indg; (aa) N Ind$ (go).

But (~1 = ‘~0 is an irreducible supercuspidal (and discrete series) representation of Ml = MF and

Indzl (al) = Ind$ (00) N Ind$ Ind$ (co)

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Thus we have irreducible supercuspidal (and discrete series) representations 01 of Mi and (~2 of M2 so that

Clearly Ml and M2 cannot be conjugate in G since Ml is connected while M2 is not. Further, the representations Indg% (ai) have a nontrivial intertwining, but are not equivalent.

A final nice property of parabolic induction in the connected case is the theory of R-groups. If P = MN is a parabolic subgroup and LT is an irreducible discrete series representation of M, then R is a subgroup of WC(O), the group of Weyl group elements fixing 0, which determines the reducibility of Indz(n). Its most basic property is that the dimension of the algebra of self-intertwining operators of Indz(g) is equal to the number of elements in R. The following example shows that there could not be such a simple R-group theory in the disconnected case for N-parabolic subgroups.

EXAMPLE 6.10. - Let Go = SL2(F) x SL2(F) = {(z, y) : z,y E SL2(F)}. Let G = Go U 7G” where y(z, ?/)y-’ = (y,~). Let B1 = A,Nl denote the usual Bore1 subgroup of G1 = SL2(F) w h ere Al is the subgroup of diagonal elements and Nl is the subgroup of upper triangular matrices with ones on the diagonal. Then B” = B1 x B1 is a Bore1 subgroup of Go and B = NG ( B”) = B” u -yB”. B” and B have Levi decompositions B” = M”N and B = MN where M” = A0 = Al x Al, M = M” U yM”, and N = Ni x Ni. We have Weyl groups W(G”,Ao) = W(G1,Al) x W(G1,Al) N 22 x 22 and W(G, A’) = W(G’, A’) U rW(G’, A’) N Ds, the dihedral group of order 8.

Let x1 be a non-trivial character of Al N FX of order two so that Indg: (xl) = x1 $ 7rTT, is a reducible principal series representation. Note that the R-group here is RI = W(G1,Al) N 2 2 and we denote the irreducible constituents of the induced representation by ~1, 7rs to indicate that they are parameterized by the trivial and sign characters p1 and ps of RI respectively. Now let x = XI @ x1. Then

where for i, j E { 1, s}, we write 7rij = n; @ 7rj. Note that 7r$ N 7rj;, so that 7r$ 21 rij if and only if i = j. Let IIIij be an irreducible representation of G such that 7rij is contained in the restriction of II,, to Go. Then if i = j we have

where Q is the non-trivial character of G/Go and IIIij 8 v $ IIij. If i # j we have

Ind$(r;j) = I& N IIji = IndEo(aji).

In this case we also have IIij @J 7 E l&. Thus we have

Note that the above decompositions are reflected in the R-groups as follows. First, the R-group for Ind$(x) is given by R ’ = RI x R1 = W(G”,Ao) z 22 x 22. It has

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4 characters pc = p; @ pj, i, j E { 1, s} corresponding to the irreducible representations 7rij. The R-group for IndgO (x) is R = W(G, AC,) = R” U rR”. By abuse of notation we will denote the non-trivial character of R/R0 by the same letter 7 as used above for the non-trivial character of G/Go and below for the non-trivial character of M/MO. The irreducible representations of R are the characters pii and pi; @ q where

and the two-dimensional irreducible representation

Thus we have the irreducible constituents of IndgO (x) parameterized by the irreducible representations of R and occurring with multiplicities given by the degrees of the corresponding representations.

Now we consider Ind$(x). Since M = M” U y&f0 and x7 = x, we have

Indgo(x) = (T $ (a 8 q)

where 0, g @ n are distinct one-dimensional unitary representations of M which restrict to x on MO. Now, using transitivity of induction and properties of tensor products, we can write

Indgo (x) E IndgInd$ (x)

pv Indz(a) $ Indg(a @ q).

Now since Indg (c @ 77) N Indg (u) @ v, we see that

Thus we can assume that IIii, II,, were chosen so that

This example exhibits a number of unpleasant features. First, we have irreducible discrete series representations cl = ~7 and (~2 = g @I n of B such that Indg(cri) and Indg(a2) have a non-trivial intertwining, but are not equivalent. Second, Indg(o) has 3 inequivalent irreducible subrepresentations, each occuring with multiplicity one, so that the dimension of its space of intertwining operators is 3. There are no subgroups of any possible Weyl groups here with order 3.

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REFERENCES

[ 11 J. ARTHUR, Unipotent automorphic representations: conjectures (Societe’ Mathkmatique de France, A&risque, Vol. 171-172, 1989, pp. 13-71).

[2] J. ARTHUR, On elliptic tempered characters (Acta Math., Vol. 171, 1993, pp. 73-138). [3] A. BOREL and N. WALLACH, Continuous Cohomology, Discrete Subgroups, and Representations of Reductive

Groups (Annals of Math. Studies, no. 94, Princeton University Press, Princeton, NJ, 1980). [4] L. CLOZEL, Characters of non-connected reductive p-adic groups (Canad. J. Math., Vol. 39, 1987, pp. 149-167). 151 S. S. GELBART and A. KNAPP, L-indistinguishability and R groups for the special linear group (Adv. in Math.,

Vol. 43, 1982, pp. 101-121). [6] D. GOLDBERG, Reducibility for non-connected p-adic groups with G” of prime index (Canad. .I. Math., Vol. 47,

1995, pp. 344-363). [7] HARISH-CHANDRA, Harmonic analysis on reductive p-adic groups (Proc. Sympos. Pure Math., Vol. 26, 1973,

pp. 167-192). [8] HARISH-CHANDRA, Harmonic analysis on real reductive groups III. The Maass-Selberg relations and rhe

Plancherel formula (Ann. of Math., (2), Vol. 104, 1976, pp. 117-201. [9] A. W. KNAPP and G. ZLJCKERMAN, Classification of irreducible tempered representations of semisimple Lie

groups, (Proc. Nat. Acad. Sci. U.S.A., Vol. 73, 1976, pp. 2178.2180. [lo] R. P. LANGLANDS, On the classi$cation of irreducible representations of real algebraic groups, in Representation

Theory and Harmonic Analysis on Semisimple Lie Groups (Mathematical Surveys and Monograph, no. 3 1, American Mathematical Society, Providence, RI, 1989, pp. 101-170).

[I 1] J. D. ROGAWSKI, Trace Puley-Wiener theorem in the twisted case (Trans. Amer. Math. Sot., Vol. 309, 1988, pp. 215-229).

[12] D. SHELSTAD, L-indistinguishability for real groups, (Math. Ann., Vol. 259, 1982, pp. 385-430). [ 131 A. J. SILBERGER, Introduction to Harmonic Analysis on Reductive p-adic Groups (Mathematical Notes, no. 23,

Princeton University Press, Princeton, NJ, 1979. [ 141 M. TADIC, Notes on representations of non-archimedean SL(n) (Pacijic J. Math., Vol. 152, 1992, pp. 375-396).

(Manuscript received December 20, 1995.)

D. GOLDBERG

Department of Mathematics, Purdue University,

West Lafayette, IN 47907.

R. HERB

Department of Mathematics, University of Maryland,

College Park, MD 20742.

4e S!?RIE - TOME 30 - 1997 - No 1

Some results on the admissible representations of non-connected reductive p-adic groups

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