Hecke Algebras and Harmonic Analysis on p-Adic GroupsAuthor(s): Mark ReederReviewed work(s):Source: American Journal of Mathematics, Vol. 119, No. 1 (Feb., 1997), pp. 225-249Published by: The Johns Hopkins University PressStable URL: http://www.jstor.org/stable/25098531 .Accessed: 02/08/2012 22:04

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HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS

By Mark Reeder

Abstract. We compute Plancherel measures and formal degrees for unipotent representations of p adic groups, using Hecke algebra isomorphisms. The results verify special cases of the conjectural

uniformity of formal degrees in an L-packet.

1. Introduction. In this paper we compute some numerical quantities aris

ing from induced representations of reductive p-adic groups, namely Plancherel

measures, formal degrees, and the ubiquitous constant called j(G/P) by Harish

Chandra. All three are given by integrals which are difficult to evaluate directly. Here they are found with the aid of Hecke algebras, in the following situations:

We determine the Plancherel measure when the inducing representation is com

pactly induced from a cuspidal unipotent representation on the points of the Levi

factor in the residue field. In passing, we note the irreducibility and intertwining criteria which follow. We then express the formal degree of the square integrable

subquotients in terms of Hecke algebra formal degrees and compute it explicitly in some cases. This method of computing formal degrees has been known for

some time, but a general proof seems not to have appeared. We include it here for

completeness and also to verify some formal degrees for SOj and Spe predicted in [Rl] by a conjecture on the uniformity of formal degrees in an L-packet. Fi

nally, we compute j(G/P) for any self dual parabolic subgroup. It turns out to

be a Poincare polynomial. The idea is to use Hecke algebra isomorphisms ([L2,3], [M]) to relate the

induced representation to a principal series module over an affine Hecke algebra

(which has unequal parameters and is not generally the Iwahori-spherical algebra of another p-adic group). To make the isomorphisms sufficiently precise is a mat

ter of checking details, for which we find it helpful to discuss the Hecke algebra and its action on the induced representation simultaneously. This is done in stages,

beginning with the compact part of the Hecke algebra (essentially Howlett-Lehrer

theory), then the commutative part, and finally the relation between the two. Then

comparison of intertwining operators leads to the Plancherel measure, up to the

constant j(G/P). The formal degree formula is proved by combining the founda

tional work of Matsumoto [Mat] on the Hecke algebra side with an argument in

Manuscript received October 4, 1995.

Research supported in part by a grant from the National Science Foundation.

American Journal of Mathematics 119 (1997), 225-249.

225

226 MARK REEDER

[BK] on the group side. We work everything out in detail for unramified classi

cal groups and maximal parabolic subgroups. In the last section j(G/P) is found

by another comparison of intertwining operators, this time between degenerate

principal series.

Howe and Moy were the first to use affine Hecke algebras to determine formal

degrees for representations of /?-adic groups, although similar ideas appeared earlier in the work of Lusztig and Howlett-Lehrer on finite reductive groups (these last play a key role here). The technique has since been used for particular groups in [HM1,2], [My 1,2], [BK], [CMS], [W].. .Recently, a categorical equivalence

has been established in [M2] and [MP], as a consequence of a general criterion

for such in [BK2], and our results may be viewed as numerical incarnations of

this equivalence.

Finally, we mention that our inducing representations do not have Whittaker

models [R3], hence the Plancherel measures determined here do not fall into the

general framework established by Shahidi in [Sh]. We introduce some notation while making things more precise: Let G be an

unramified connected reductive group over a nonarchimedean local field F, with

maximal F-split torus S. We put G = G(F), S = S(F), and likewise for other

F-groups. Let A, ? be the roots and a set of simple roots of S in G. We take a

subset Jci which is self-opposed in every overset of simple affine roots (see

below). To J corresponds a parabolic subgroup P = Py and a parahoric subgroup H = Hj. Let L be the reductive quotient of P. It is the centralizer of the subtorus

A c S cut out by the kernels of the roots in J. Let M be the reductive quotient of H. Then M is the rational points of a reductive group over the residue field

of F and is also the homomorphic image of a special maximal compact modulo

center subgroup H? of L. Let a be a cuspidal unipotent representation of M9 and

let 6 denote its contragredient. The inflation of o to H? is trivial on A fl H?9 so

for any unramified character r of A we can define the representation rcr of AH?9

and compactly induce to get a supercuspidal representation

[a]T := ind^o (ra)

of L. Now inflate [o]r to P and induce (unitary to unitary) to G to obtain

I(T9o):=lnd$[a]r.

Consider the algebra H(o) of locally constant compactly supported End (<j)

valued functions C on G satisfying C(hxgh2) =

<?(hx)C(g)<j(h2) for A/ G H, g G G.

The multiplication in H(a) is given by CC'(g) = jG C(x)C(x~xg) dx, where dx is

the bi-invariant measure on G for which H has volume one. For any admissible

representation n of G, we have an action tt of H(a) on the space of embeddings

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 227

Horn// (a, IT) defined by

(7r(C)ip)v= I Y\(x)il)(C(xyv)dx,

where C G H(o), i\) G Horn// (o, II), v eo, and C(j<ris the endomorphism of o

induced by C(x). When n = /(r, cr), F[(jc) is right translation by x and we denote

the corresponding H(o) action by 7ra.

It follows from ([L2,3], [M]) that H(o) is isomorphic to an affine Hecke

algebra, although the isomorphism is not quite uniquely determined. This is to

our advantage, as we can choose the isomorphism to make some unpleasant ?

signs disappear. We then show that Horn// (o,I(t, o)) is a principal series module

over H(a), and the space of intertwining operators between two such induced

G-modules is isomorphic to the corresponding space of H(o)-module interwiners,

about which much is known (cf. [Cas2], [R2]). For example, the intertwining map

I(t, o) ?

I(wot, a) (wo is the longest element in a certain Weyl group) restricts

to a scalar multiple of the standard intertwiner between Hecke modules. This

scalar turns out to be independent of r, and leads to the formula for Plancherel

measure on this series of representations. Using Kato's reducibility critera for

Hecke algebras [K], we get the reducibility criteria for I(t,o). Using results

in [R2], we deduce that all intertwiners between I(t,cj) and I(wt,o) are given

by residues of standard intertwiners, and we give the non-intertwining criteria.

Finally, formal degrees of square integrable subquotients of /(r, o) are expressed in terms of formal degrees for W(a)-modules. The latter are not yet known for

Hecke algebras with unequal parameters (which are usually what arise here),

except in type Ax, where they have been determined by Matsumoto [Mat]. For

the moment, this restricts our explicit calculations of formal degrees to maximal

parabolics P, and a few other cases.

2. The compact parts of H(o). Much of the structure of H(o) and its

action on Horn//(a,/(r, cr)) do not require calculations beyond those of [HL], Let Ao and So be the maximal compact subgroups of A and S. The quotient is

a lattice Ay = A/Ao contained in the lattice A =

S/So- Let W be the (spherical)

Weyl group of S in G, and let W(J) =

{w G W: wJ = /}.

Let ia = __. U c_o be the simple affine roots. For each a G Zfl

? /, there is a

maximal parahoric subgroup K& containing H. We have K& = HWV^)H, where

W?(&) is the stabilizer in WA of that vertex v(a) G R ? A which is orthogonal to

the roots in I - a and lies in the closure of the alcove determined by Ifl. The

special maximal parahoric Ko := K&0 has the property that G = PKo.

We assume that / is strongly self-opposed, in the following sense. For every

proper subset / c Sfl, let w7 be the longest element in the (finite) subgroup of

WA generated by reflections from /. Set

ra =

VV/U(iWy.

228 MARK REEDER

(The notation anticipates a later definition of a.) Our assumption is that r^ = 1,

or equivalently, that raJ = /, and we require this to hold for every a G Za ? J.

It automatically holds if the Dynkin diagram of J is unique of its type in that of

Zfl, and many other cases as well.

It is shown in [Car, 10.3.1] that there is a subgroup AfjCKo containing H?

as normal subgroup with quotient W(J). Let w G A/y represent w G W(J)9 and

let A G A represent A G Ay. Now o is invariant under rational automorphisms of

M (see [DM, 13.20] and the proof of (5.1) below). Moreover, A centralizes H?.

Hence, for each vv and A, we have an intertwining operator

p(w\) =

p(w): wo ? o,

that is, o(h) o p(w) =

p(w) o a(w~xhw). The group G is the disjoint union G =

]J Hw\H9 over all A G Ay and w eW

such that w and w~x send J to positive roots. The space of functions in H(o)

supported on a given double coset HwXH is at most one dimensional. If it is

nonzero, the cuspidality of a forces w G W(J)9 A G Ay (cf. [M, 4.13]). Hence

Lemma 2.1. H(cr) has a linear basis {Cw\: w G W(J)9 A G Ay}, where

Cw\(w\) =

p(w).

Let K be a parahoric subgroup containing H9 and let H(K9 a) be the subal

gebra of functions in H(cr) with support in K. As a linear space, H(K9 a) is finite

dimensional, since K is compact. The algebra structure of H(K, a) is unchanged if H and K are replaced by their finite reductive quotients, and was determined

in the latter context by Howlett and Lehrer [HL] up to a parameter, which was

then computed by Lusztig. The algebras H(K, a) are described as follows. It suffices to consider maximal

parahorics, so fix a simple affine root /? G Xa ? J. Let v =

v((3). Let WV(J) =

{vv G Wv: wJ = /}. Then WV(J) is a finite Coxeter group with simple reflections

rQ, for a G ta -

(J U {/?}). The algebra H(KV, a) is the Hecke algebra of WV(J),

with generators Tra, a G JLa ?

(J U {/?}), satisfying the usual relations, with

parameters qa > 1. These last are obtained as follows. Let Kv be the finite

reductive quotient of Kv. The image of H under this quotient is a parabolic

subgroup of Kv corresponding to the subset J of the simple roots of Kv. Let

H(a) be the preimage in Kv of the parabolic subgroup corresponding to the

subset J U {a}. Then qa is the ratio of the dimensions of the two consitituents

of the induced representation Ind^(a) o. These ratios were computed by Lusztig

when a is cuspidal unipotent, and may be found in [Car, p. 464].

Lemma 2.2. When K = Ko is a maximal special parahoric subgroup, the re

striction ofthe module Horn// (cr, I(r, a)) to H(K, a) is isomorphic to the left regular

representation ofthe latter.

Proof. Let Y be the kernel K ?> K. Then Y is contained in the kernel

of H ? M. The Iwasawa decomposition implies that as A'-modules, we have

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 229

I(t,cj)t c__ Ind//_r via the map/ ?- /_, where fx(k) =f(k)(l). Hence, as H(K,o)

modules, we have

Horn// (o, I(t, cr)) -

Horn// (o, Ind? o) -U H(K, cf),

where (iil))(k)v =

ip(v)(k), for k G K, v G a.

The left regular action of W^, o) on itself is isomorphic to the left action of

H(K,o) on H(K,d) given by C D = DC, where C(g) =

C(g-1)^ Finally, one

checks that i defined above is an H(K, cr) module map.

3. A basis of Horn// (cr, /(r, cr)). Let L? be the subgroup of L on which

|x| = 1 for every rational character \ of L. Then H? is maximal compact in

L? and we set [cr] =

ind^ocr. Identify [cr]T = [a], via restriction to L?, and

consider the following functions therein. For w G W(J), h G H, v G a, we define

ipwAv ? M to have support in //?, and for A: G H?,

^w,h,v(k) =

cr(k)p(w)o(h)v.

Then define ipw,v G /(r, cr) to have support in PwH, such that

ipwApwh) =

t82(p)Rp^Xv,

where p e P, h e H, 8 is the modular function of P and Pp is right translation

by p. One checks that Vy t> is well-defined, independent of the representative of

w, and satisfies

ipwAw)(h) =

?(h)p(w)v, 1pw,*(h)v =

Phipw,v

In particular the map v i? VVt> *s an -^-module embedding

ipw: a ?> I(t,o).

Lemma 3.1. The set {ipw: W(J)} is a basis of Horn// (cr, /(r, cr)). Under the

isomorphism of Lemma 2.2, ipw G Horn// (cr,/(r, cr)) corresponds to Cw-\.

Proof The ^w are linearly independent by consideration of supports. Let K,

r be as in the proof of Lemma 2.2, and let bars denote image in K/Y. We have

/(r, af ~

Indg o ~ Ind??a.

The result follows from Frobenius reciprocity and the finite group intertwining

number, (cf. [Car, 9.2.4]). The last assertion is obtained by tracing the definitions.

230 MARK REEDER

Let wo be the longest element of the reflection group W(J). We have wo =

vv^wy, with notation as in ?2. Combining Lemma 2.2 and Lemma 3.1 we obtain

Corollary 3.2. ipWo generates the H(o)-module Horn// (o, I(r, cr)).

4. The commutative part of H(a). Let N be the unipotent radical of P,

H^=HDN, and for A G Ay, set H\ =

A//JA-1. Let AJ be the set of A G Ay for

which H\ C //J.

Lemma 4.1. For A, p G Ay, we have C\C^ =

C\+^.

Proof. We have

CxC^(g)= Y C^X-'h-'g).

For 7 G Ay, one checks that Hq f) XHpH^~x =

H\ if 7 = A + p and is empty otherwise. For vv G W(J), we find that Hq D XHpHj~xw

= 0 always. The result

follows.

One consequence of [M, 7.12] that can be seen without further calculation is

that any element of H(cr) supported on a single H ? H double coset is invertible.

In particular, each C\ is invertible. Let C[T] denote the subalgebra of H(o)

generated by all C\ and their inverses, for A G Ay. Here T = Cx (g> Ay is the

complex torus of unramified characters of A, and as the notation suggests, this

subalgebra turns out to be the coordinate algebra of T.

Proposition 4.2. Let r G T, and let tpWo G Horn// (cr, I(r, cr)) be as in (3.2). Then for X G Ay, we have

TT^Cx^wo =

82t(w0\wq x)tpWQ.

Thus ipWo is an eigenvector for C[T].

Proof. By (3.1), we have TTa(Cx)ipwo =

*52weW(j) awipw for some aw G C On

the other hand, the definition of ttg says

(TTa(Cx)^)V= Y RhXlpw0(?(h~l)v) =

Y RhXh~^wQ,v, H/H^H H+/H+

for all v G a. Evaluating at x, for x G W(J), we find

axipx,x,v= Y VW'C^AA-1).

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 231

This last can only be nonzero if xh\h~x G PwoH = PwoHq. The Bruhat decom

position then forces x = wo and h G H^. Now we have

aWoipw0,\,v =

VWzXwoA) =

82T(wo\wo~l)ipw0,hv,

proving the claim.

We sketch a justification of the notation C[T]. Let Y be the variety of maximal

ideals of the commutative algebra C[T]. We have just shown that TC7, Since

Ay is generated by \L ?

J\ = dimT elements A G Ay, we have a surjective map

from the coordinate ring of T to C[T], which is the inverse of restriction from Y

to T. Hence T = Y. This identification of algebras is also a consequence of [Ll], as recalled in the next section.

5. The complete picture of H(cr). We show how the compact and com

mutative parts of H(cr) combine to give the full description of the algebra and

its action on Hom//(cr,/(r,cr)). It is known that H(o) is an affine Hecke alge bra. This is sketched in [L2,3] and a complete proof is found in [M], where the

method of Howlett-Lehrer [HL] in the finite group case is extended to p-adic

groups. However, to arrive at the precise form of the result that we require (more than generators and relations), several details must still be checked.

The subgroup W(J)a C W(J)Aj generated by {ra: a G Xa ?

J} is a Coxeter

group on these generators. Let ? denote its length function. W(J)a is a normal

subgroup of W(J)Aj, and we have a semidirect product W(J)Aj = Q tx W(J)a, where Q is abelian. The length function extends to W(J)Aj as a bi-invariant

function under Qy. Let q = {qa: d _Ifl- /}, with the qa as in ?2. The Hecke

algebra H(J,q) has the C-basis {Tw: w G W(J)Aj} with relations TxTy =

Txy if

?(xy) = ?(x) + ?(y), and

T2a =

(qa -

\)Tra +qa. We call this the "IM-presentation" after [IM].

There is also a twisted tensor product presentation, due to Bernstein-Zelevinsky and Lusztig [Ll], which after invoking [HL], says

H(J,q)~H(Ko,a)?C[T],

where the algebras on the right side are defined in ?2 and ?4. To describe the

multiplicative relation between the two we must give the relevant root datum.

The importance of the following recipe will later be seen when we compute Plancherel measures.

Let AJ be the set of a G A which do not vanish on A, such that {a} U J is

contained in a base of the root system A. Because of our hypothesis on J, the

restrictions to Ay of roots in AJ form a (possibly nonreduced) root system, with

base consisting of the restrictions of roots in __. - 7 [Car, 10.10.1]. However, this

232 MARK REEDER

root system will not be part of our root datum. Instead, each root in the root

datum will be proportional to such a restriction.

We have vector spaces E = R ? Ay and E* = R <g> Y9 where Y is the rational

character group of A. As the notation suggests, these two vector spaces are in

duality, since Ay is naturally isomorphic to the group of one parameter subgroups of A. Now a G AJ induces a nonzero linear functional a on ? whose value at

A G Ay is (A, a) := val(a(X)). Let (, ) be a vV(/)-invariant inner product on E. For an affine hyperplane H C

E (not to be confused with the parahoric subgroup H)9 let ru be the corresponding

orthogonal reflection in E about H. The group W(J)Ay acts on E by affine motions

and for a G A*7, ra acts via r//a, where Ha - ker a [Car, 10.4.2]. Let H be the set

of hyperplanes H for which rn belongs to the transformation group on E induced

by the subgroup W(J)a C W(J)Aj. Let Ha be the collection of hyperplanes in

H which are parallel to Ha. The intersections of hyperplanes in Ha with the

orthogonal complement H# ~ R form a discrete subgroup Ya of H^. Let H+

be the unique hyperplane in HQ on which a is positive, such that the point H+ fl H^ generates Ya. Define a G H^ by the condition H+ n H^

= {\a}, and

set Ay = {a: a G A*7}. As translation by a is rn^ra G W(J)Aj9 we have Ay C Ay.

We also define, again for a G AJ9 the element d = 2(a,a)~xa G E*. In

other words, d = 1 is to be the hyperplane H+. According to [Bour, p. 178], the

collection Ay = {a: a G Ay} is a reduced root system with Weyl group W(J).

The inner product (, ) induces an isomorphism/: E* ? E which maps Ay to a

root system in E whose dual is Ay, so Ay is a reduced root system as well. To see

that we have a root datum, it remains only to check that Ay c Y. It is equivalent to check that (Ay,d) C Z for every d G Ay. If A G Ay, the hyperplane Ha + A

belongs to HQ, so there is m G Z such that Ha + A = Ha + a, by the definition of

a. On the other hand, Ha + X = Ha + tj^o:,

so 2(A, a)(a, a)~x = m. Finally, one

checks that/(d) = 2(a, a)~xa, so (A, d)

= (A,/(d))

= m. The root system Ay has

a base Zy = {a: d G 2-/}, and we let Ay denote the corresponding positive roots

in Ay. We now have a root datum (Ay, Ay, Y, Ay,Zy), which is always irreducible

in the cases at hand.

For A G Ay, we have a polynomial function ex G C[T] given by ex(z ? x) =

z(A,x) jn particular, for each a G Ay, we have a polynomial function ea. Define

rational functions Ca on T as follows: In all but one case, we have

>. __ <?a ^a

The exceptional case is when Ay has type Bn, n > 2, and a is a short root. Then

i i i _i

. _ (ffaffao ~

ea)(ea + ffaffap2 )

1 -^2a

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 233

where ao is the highest root in Bn. Note that Ca = q<*Ca, where ca is the usual

c-function (cf. [Cas3]). Now the relation between the two factors in H(J, q) is given, for a G ?y,

9 G QT], by

0(Tra ~

qa) =

(Tra -

q?W* + (0r? -

OKa,

where the action of W(J) on C[T] is induced by its action on Ay. It turns out that H(J,q) is isomorphic to H(o). In fact, we have already

seen how the algebras H(Ko, cr) and C[T] are subalgebras of H(o). That they

generate H(o) and have the correct multiplication relation follows from the IM

presentation for H(o) given in [M] and its translation into the tensor product formuation in [Ll], once we check that the cocycles appearing in [M, 7.12] are

trivial. This follows from

Lemma 5.1. The representation o extends to the normalizer N(H) ofH in G.

Proof. Each n G N(H) induces, by conjugation, an automorphism of M which

extends to the algebraic group M over ?q in which M is the fixed points of a

Frobenius morphism / commuting with n. The group M is quasi-split, hence

admits an /-stable splitting. Let T be the group of automorphisms of M which

preserve the splitting. Then T is isomorphic to a subgroup of the automorphism

group of the Dynkin diagram of / and there is a semidirect product Aut (M) =

Int(M)r, hence a semidirect product Aut(M)f =

InUM/r* There is a natural

map M ? Int(M)f, and a factors through a cuspidal unipotent representation ox

of Int(M)f (cf. [DM, 13.20]). Moreover, [loc. cit.] it is known that ox ~

d[ for

any 7 G T, and we choose such isomorphisms /_7 G End(cri). We have assumed

J to be contained in the simple spherical roots. The list of groups with diagram J admitting a cuspidal unipotent representation [Ll] forces T to be cyclic or 53. It is now easy to find scalars c7 so that 7 1?

c1R1 is a representation of T on ax, and since /_7 intertwines the Int (M)f-action, this extends ox to Aut(M)f.

Finally, invoking [Ll], [M] we have

Proposition 5.2. We have an algebra isomorphism jo'. H(J, q) -=> H(a) which

sends Tw, for w G W(J)Aj, to a scalar multiple of CWf and is the identity on the

subalgebras H(Ko, cr) and C[T].

6. Normalizing the Hecke algebra isomorphism. There is some ambigu

ity in the scalars in Proposition 5.2, due to support preserving automorphisms of H(J, q) and undetermined signs. We show how to choose the isomorphism in

Proposition 5.2 so that one cancels the other.

For w G W(J)Aj, let bw G C* be defined by the relation jo(Tw) = bwCw,

according to Proposition 5.2. The isomorphism jo is not uniquely determined.

234 MARK REEDER

However, for A G Ay, [M, (7.10)] implies that we may choose yo so that

bx = ex(qx[Ho #a1)1/2>

where ex = ? 1 and for any vv G W(J)Aj, qw is defined as follows. If vv = urx r* is a reduced expression with r; =

ra/ simple reflections in the Coxeter group W(J)a and wGd, then qw =

qa\ ?-

qak> Since the cocycle /x in [M, 7.12] is trivial, and

p(wv) =

p(vv) for vv G vV(/) and i/ G Ay, it follows from [M, (7.11)] that the map v I? ev is invariant under conjugation by W(7).

Now, in H(J,q), we have [Ll, 3.3]

-1/2 *a = qx T\>

for A G Ay. Hence

jo(6x) = 6A[//J : ^])1/2CA

= ex6L2(X)Cx.

It then follows from Proposition 4.2 that there is an unramified character %o ? T

such that xo(A) = ex for A G Ay. Note that \o

= 1' anc* Xo is W(/)-invariant, since eA is invariant. Hence the map Tw (g) Ox ?-^ Xo(A)Tw ? ^a? for vv G W(/), A G Ay, is an algebra automorphism of H(J9q). Composing this automorphism with jo gives a possibly new Hecke algebra isomorphism

j: H(J9q)?+H(cj),

which we use from now on. This new isomorphism has the following property. For r G T, let f = wqt. Now Proposition 4.2 says

j(exWW0 =

f(\WW0

We have now arrived at the main point of this section. For r G T, let CV be the

one dimensional C-vector space on which C[T] acts by evaluation at r.

Proposition 6.1. Via the algebra isomorphism j: H(J,q) ?

7Y(cr) defined

above, the H(a) module Horn// (cr, I(r, a)) becomes the principal series module

M(f):=H(J,q)?C[T]Cr.

Moreover, ip*Q G Horn// (a,I(r,o)) corresponds to vf := 1 ? 1 G M(f).

Proof. By the above interpretation of Proposition 4.2, there is a map M(f) ?>

Honif/ (cr,/(r, (cr)) sending 1 ? 1 to ^0.

This is surjective by Corollary 3.2, and

Lemma 3.1 implies that both sides have dimension | W(J)\.

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 235

This implies a reducibility criterion for /(r, o). For general Hecke algebras, the reducibility of the principal series was determined by Kato. Consider the

rational function C =

Il/3eA+ C/3> w^h C/3 as in ?5. Combining Proposition 6.1 and

[K, Theorem 2.2], we obtain

Corollary 6.2. /(r, cj) is irreducible if and only if the following both hold.

(1) CO-)C(f)7*o.

(2) The stabilizer in W(J) ofT is generated by the rafor which Ca has a pole atT.

7. Intertwining operators. We now relate the intertwining algebra of I(t, o) to that of its corresponding Hecke module. The next result follows from an equiv alence of categories ([BK2], [MP]), but it is easy to prove the part we need.

Proposition 7.1. The functor Horn// (cr, ) induces, for every r G T and w G

W(J\ an isomorphism

Homo (/(r, cr), I(wt, cj)) c__ Hom^(y,q) (.M(w>or), M(wowt)).

Proof. Let U = ker(_? ?

Af). In view of Proposition 6.1, it suffices by

[Cas] to show that every submodule V of /(r, cr) contains o in its [/-invariants.

Let Vn denote the Jacquet module with respect to N. It follows from Frobenius

reciprocity for smooth induction that [cr] ? r_>2 is a quotient of the L-module

V)v- Hence [cx] <g> t~x8~2 is a submodule of the contragredient (V#)v. Now by Frobenius reciprocity for compact induction, we have

0 j HomLn77 (cr, (VN)y) = HomLn77 ((VN)Lnu, o).

On the other hand, U admits an Iwahori factorization with respect to L, so [BD,

3.5.2] implies that the natural map Vu ? (V/v)Lnf/ is surjective. It follows that

o appears in Vu.

The standard intertwining map A^: I(t,o) ?> /(wr,cr) is defined formally

by the integral

-4Xs) = / p(w)~lip(w~lng)dn, JN

where _V is the unipotent radical of P, p(vv)-1 acts on [cr] by acting on the value

of functions, and Hq = NHKo is given volume one. The integral converges for r

in an open subset of T. By restricting to Ko, all /(r, cr)'s can be represented on the

same space and for fixed ip and g, the function r ?? A^ip(g) extends to a rational

function on T. For generic r, it is known that _4?. spans Home (/(r, cr), I(wt, cj)). There are likewise "standard" intertwining operators between principal series

modules for Hecke modules (see below) and for generic r they span

236 MARK REEDER

Hom/ft(y,q) (M(wot), M(wowt)). It follows that standard intertwiners correspond,

up to scalar, under the isomorphism in Proposition 7.1. In fact, for arbitrary r, it

is shown in [R2] that their holomorphic linear combinations span the intertwining

space. Let us make this more precise, and at the same time give the application to I(t, a). Let W(J, r) be the stabilizer of r in W(J). Then, combining [R2, (6.3)] and Proposition 7.1 above, we have

Corollary 7.2. Given ro G T with stabilizer W(J, ro) in W(J), vv G W(J) and

an intertwining operator A: /(ro, cr) ?

/(wro, cr), there are rational functions

fx on T, indexed by x G vvW(J9 ro), with the property that ^2xewW(j,T0)fx(T)^x

*s

holomorphic at ro and when evaluated at ro becomes A.

By [R2, (14.7)], we also have an intertwining analogue of the reducibility criterion in Corollary 6.2. For vv G W(J)9 let rjw be the product over the roots

/? G Ay made negative by vv, of the numerators of the Qp9 as given in ?5.

Corollary 7.3. /(r, o) admits no nonscalar endomorphisms if and only if

Corollary 6.2(2) holds, and rjw(r) ^ Ofor every vv G W(J, r).

8. Plancherel measure. Let us consider in particular the standard intertwin

ing map A?0: I(r,a) ?> I(f, a), which is used in the calculation of Plancherel

measure. If xjj G I(r,a) has support in the big cell PwoN, the integral itself

extends to a rational function. The composition

A^Q o

Al0: I(r, a) ?+ I(f, a) ? I(r, a)

is a scalar c(r) times the identity. Recall (cf. [Sh]) that the Plancherel measure

p(r, a) is given by

/i(r,a) =

7(G/F)2c(r)-1,

where 7(G/P) is a constant independent of r and a. In this section we compute

c(r), and j(G/P) is found in ? 11 below. Recall the rational function (

= ]1/?ea+ C/9> where Ay is the positive system

defined in ?5, and the C/3 are the rational functions appearing in the relations for

H(J,q).

Proposition 8.1. For generic r G T we have

C(T) = C(r)C(r) Qw0[Hq : H\]

Proof. We have an induced map A?0: Horn// (cr, I(r, a)) ?

Horn// (cr, I(f, a))

and

xew(j)

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 237

for certain ax G C. For v G o we have, using the Bruhat decomposition,

axv = 0_^i,ih,i;(U)

= ^0Co,^g)(U)

= p(vvo)-1 / ^wo,i;(wo?)(1l)^

= pO^o)-1 / ipw0Av(lL)dn = p(w0)~l / /o(vv0)(f)cin = ^

Jh* Jh+

Thus ai = 1.

On the other hand, the Hecke module M(f) = H(J, q) ?qt] Cf admits, for

generic r, an intertwining operator F: M(f) ?

M(r), unique up to scalar. It

is enough to define its value on vf = 1 ? 1 G A1(f). We require some preliminary remarks. Let To be the largest open set in T on which the Ca are holomorphic and nonzero for all a G Ay. Note that To is W(J) stable, so W(J) acts on the

field C(To) of rational functions on To. For f G To, the H(J, q)-action on M(f) extends to an action of the localized algebra H(Jo)

= r^(7,q)(g>C(To). The

multiplication formulas in this algebra are the same as H(J, q), except now each

Ca belongs to W(To), and the fundamental relation simplifies to 0(Tra ?

qa) =

(Tra -

qa)6r? for 6 G C(T0), a G Xy.

Choose a reduced expression in W(J) for the longest element wo = rn rx,

where r, = ra/ for some a, G Xy. In H(To), consider the element

Fw0 =

(Trn ?

qan + Can) ' ' '

(Tr\ ~

^a, + Ca,)

The fundamental relation in rt(To) and Frobenius reciprocity imply that the map

vf ?

FWQvT extends to an intertwining map F: M(f) ?

M(t). It is easy to

see that any such intertwiner is a scalar multiple of F (remember that f G To). Likewise FWo defines an intertwiner F in the opposite direction, and a direct

computation shows that

FoF = C(f)C(r)/.

The map A^ on Horn// (cr,/(r, cr)) must be, via the isomorphism j of Propo

sition 6.1, a scalar multiple of F, say ^A1^Q = F. Apply both sides of this equation

to VC0

and look at the coefficient of ty\ in the result. We have seen that on the

left side it is k. Expand FWQ =

T,xew(j) Tx ? 0X, with 0X G C(T0). Note that

0WQ = 1. Put jTWo

= 6Ch;0. For a linear combination (2 of Tx or ^>J, let [Q]x be

the coefficient of T\ or ^[. We have

K Y a^= Y ex(T)7la(jTxWW0. xew(j) xew(j)

Using Lemma 3.1 and the definition of b, we have

IMyW^h = KyTjc^]! =

b-x[TxTW0]x.

238 MARK REEDER

It is known [Car, 10.9.1] that Tx appears in TxTy if and only if x = y~~x, and then

it appears with coefficient qy. It follows that k = qWob~x.

It follows from [Car, 10.8.3] that b = ujql{2[H^

: H\]~x/2 for some root of

unity uj. In the proof of [loc. cit.], we find that uj is, up to sign, the products of values of a cocycle which we have shown in Lemma 5.1 to be trivial. Hence

uj = ?1 and the proof of (8.1) is complete.

9. Formal degrees. We turn now to the connection between Hecke algebra

isomorphisms and formal degrees. The basic idea goes back to [HM1] for GLn,

but the proof in the form we require does not seem to have been written in one

place. To this end, we shall cobble together [Mat] and [BK, Chapter 7]. Though

expressed in terms of GLn, the arguments in the latter are completely general. Let ( | ) be a positive definite hermitian form on a, invariant under H.

This defines an involution " ""

on End (a) by (u,Av) = (Au, v) for u,vea and

A G End (cr), and thus induces an involutive anti-automorphism on H(o) given by

C(g) =

C(g~x), which finally leads to a positive definite hermitian form A on 7i(o)

given by h(C,D) = trace[CD(lG)]. Let H2(cr) be the corresponding completion. We need to compare this hermitian space with its analogue for H(J, q).

We also have an anti-involution on H(J, q) given on the IM basis by Tw =

Tw-1, with corresponding hermitian form h'(T, S) = TS( 1), and completion H2(J, q),

on which H(J,q) acts by left multiplication. A finite dimensional simple left

H(J, q)-module tt is "square integrable" if it may be realized as an H(J, q) sub

module Vn of H2(J, q). According to [Mat, (2.4.3)], given X, Y G Vn, the element

(matrix coefficient)

XY:= Y q^h'(X,TwY)Tw wew(j)Aj

belongs to H2(J, q). (Even though H2(J, q) is not an algebra, we use this notation

because the same formula defines the product in H(J, q).) Moreover, there is a

positive real number d(n) ("formal degree"), depending only on the isomorphism class of tt, satisfying

h,(XxYx,X2Y2) =

d(TT)-xh'(Xx,X2)h'(Yx,Y2),

forallX,,X2,Fi,y2 G Vv.

For example if dimT = 1, Matsumura [Mat, p. 47] has computed the d(7r)'s

in case ?2=1, and it is easy to modify his result for nontrivial finite ?2. Let

the parameter set be q = {qa,qo}- If qa

= <7o> there are |Q| square integrable

H(J, q)-modules \ ? ns, one for each character x of ?2. They all have the same

formal degree

d(X?TTS) = \n\-l^?. qa + 1

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 239

If qa ^ qo we must have \Q\ = 1, and in addition to irs there is another square

integrable module no. The formal degrees are given by

,, . ffaffo ~ 1 Mnr v \qa

~ go I

Our algebra isomorphism _/: W(7, q) ?> W(cr) is support preserving, and

thus automatically respects the involutions on both sides (otherwise, it is easy to

find a nontrivial automorphism of H(J, q) sending each Tw to a scalar multiple of

itself, an impossibility by the IM relations, see [BK, p. 194]). Using the support condition again, it is straightforward to check that

h(j(T),j(S)) = dim(o)hf(T,S),

for T, S G H(J, q). In particular, j extends to the completions, where the above

relation between hermitian forms remains valid.

Now let n be an irreducible square integrable representation of G containing a upon restriction to H, and let tt be the simple H(J, q)-module corresponding to Horn// (cr, FI) via/ Then we may realize n ___ Vn c H2(J, q) as above, and by the argument in [BK, Chapter 7], we have

h(j(XxYx),j(X2Y2)) = d(Y\)-xh(j(Xx),j(Xi))h(j(Yx),j(Yi)),

for all Xx,Xi,Yx,Yi G Vn, where now _/(Il) is the formal degree of I~I, with Haar

measure assigning volume one to H.

Putting everything together and recalling how formal degree varies with Haar

measure, we find

Proposition 9.1. Ifllisa square integrable irreducible representation ofG and 7r = Horn// (cr, 11) is the corresponding H(J, q) module, then

mti\ dim(o)d(7r) d(U)=

vol(ff) *

10. Examples. We illustrate our formulas for formal degrees and Plancherel measures when the group G = G? is an unramified classical group of the form

SC_-+i, SOfr, SU?, or Sp2n- In each case, the bilinear form on the vector space

Vn is determined by a matrix which we take to be

" 0 ... 0 1

"

0 ... 1 0 Q= . . .

1 ... 0 0

240 MARK REEDER

for the orthogonal and unitary groups, and Q' = ~r for Sp2n- The only

maximal parabolic whose Levi subgroup admits a cuspidal unipotent representa tion is the stabilizer P of an isotropic line. The Hecke algebra H(J9 q) is of type

Ai and the parameter set is q = {qa,qa0}- These, along with the root datum are

determined as follows.

We use the recipe given in ?5 to determine the root data. Let w be a uni

formizing parameter of F. In all cases, Ay is generated by the (coset of) the

diagonal matrix A := diag(cc7,1,..., l,w~x). Let e G E* take value one on A,

and for s G C, let rs G T be the unramified character of A given by rs(X) =

q~s. For the orthogonal groups, the affine roots in ?a

? J restrict to e and ?e+l on

E. Hence a = 2X, so |?2| = 2 and ea(rs) =

rs(a) =

q~2s. For the unitary and

symplectic groups, the affine roots in ta ? J restrict to e and ? 2e+1 on E. Hence

a = A, so ?2 = 1 and ea(rs) -

rs(a) = q~s.

We next find the parameter set, as in ?2. The choice of form defines an O

structure on Vn, hence on Gn, such that Ko = Gn(0). Let Kx be the other maximal

parahoric subgroup containing H. Letting bars denote reduction modulo the pro

unipotent radical, we have Ko ? Gn(?q). For orthogonal groups, Kx

~ Gn(?q) as

well, for unitary groups we have Kx ~

SU2(?q) x

SUn-2(?q), and for symplectic

groups we have Kx ~

Sp2(?q) x

Sp2n-2(?q). Note that SU2(?q) ~

Sp2(?q) ~

SL2(?q). This clarifies the notation SUm(?q)9 and shows that in the unitary and

symplectic cases, the parameter qao is the ratio of degrees of components of the

principal series for SL2(?q), namely, qao = q. For the orthogonal groups, we have

qa0 = qa- Now in order to have a cuspidal unipotent representation o on H?, we

must have in each case a specific form for n, and then a is unique. We list the

possible n along with the corresponding parameters, taken from [Car, p. 464]. Unless otherwise noted, k denotes an arbitrary nonnegative integer.

S02n+\: n = k2 + k + 1, qa = q2k+x

= qao

S02n' n = k2 + 1 (k even), qa = q2k

= qao

SUn: 2n = k2 + k + 4, qa =

q2k+x, qao =

q

Sp2n' n = k2 + k + 1, qa = q2k+x, qao

= q.

It is convenient to choose the Haar measure giving volume one to an Iwahori

subgroup I of Gn(F). Then the volume of H is the cardinality of the finite flag

variety H/I. The dimension of a is given in [Car], and using Propositions 8.1

and 9.1, we obtain the following formal degrees and Plancherel measures.

For S02n+\ with n = k2 + k + 1 we have |?2| =2, hence two square integrable

representations YL? containing a upon restriction to H, corresponding to ?tt as

in Proposition 9.1. They each have formal degree (with vol(X) =1)

{ ?} 2M(q2M + 0(<7+ U2*(<72 + l)2*-1 (.q2k+ D'

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 241

When k = 1, this agrees with the prediction in [Rl, p. 488], although due to a

misprint, the last term in the denominator there and in d(TX) and dfo) of [Rl,

7.4] should be (q + l)2 instead of (q + 1). For k = 1, the representations Tl? are

the only non-J-spherical unipotent discrete series representations of SO-j(F). As for the Plancherel measure, we find (see Proposition 8.1)

_ 72W#o : ?T1 _ l2q2(k+l)\q2s -

Dig'2' -

1) MT5,(T)"

araK(rrl) "

(q2s -

q2M)(q~2s -

q2k+X) '

We give a general formula for the constant factor 7 = 7(G/P) in the next section.

For SOin with n = k2 + 1, k even, there are again two square integrable

representations Il?, with formal degree

{ ?} ' 2V* + l)(q + l)2*-1^2 + 1)2*~2 (tf2*-1 + 1)'

The Plancherel measure for this series of S02? representations is

_ ^q^M){q2s _

1)(,-2, _

j) /x(T"CT)-

{q2s -

q2k){q~2s -

q2k)

'

For 5t/? with 2n = k2 + k + 4, there are two (now essentially different) square

integrable representations Us and I_o, corresponding to ns and 7ro in ?9, with

formal degrees

, n = g|(*2-')(g2^2

- !)(<? + !)(<?

- 1) (q

- ( -

l)"-2) 1 5'

(^+1)(^2*+1+ l)(^-1 + l)(^2*-3 + l)2--(^+l)*'

= gt**2-V*+1 -

gXg + l)(g -

1) (q -

( -

l)"-2) ^ ?;

(^ + l)(q2k+l + 1)(92*-' + 1)(^2*-3 + l)2 (q + l)k '

The Plancherel measure for this series of SUn representations is

M "

(?* -

0*+1)(tf-* -

qM)(qs + qk){q~s + qk)'

For Sp2n with n = k2 + k + 1, there are again two (essentially different) square

integrable representations Tls and _Io, corresponding to ws and 7ro in ?9, with

formal degrees

( s) 2*(tf2*+1 + l)(q+l){q2k + 1 ){q2k~> + 1 )2 (q + 1 )2k'

242 MARK REEDER

q^mk-X){q2k+X_q){q_xf+k 1 0) 2k(q2k+x + l)(q + l)(q2k + l)^"1 + l)2 (q + l)2*'

The Plancherel measure for this series of Sp2n representations is

12q2(k+\?(q2s_l){q-2s_l)

(qs -

qk+x)(q~s ~

qk+l)(qs + qk)(q~s + qk)'

For fc = 1, the representations Hs, Ylo are the only non-J spherical unipotent discrete series representations of Spe(F). Their formal degrees again confirm the

conjecture in [Rl].

11. The constant 7(G/P). In this section we have slightly different nota

tion. H = T is an Iwahori subgroup, o is the trivial representation of H, A = 5 is

a maximal F-split torus and T is the complex torus of unramified characters of A.

Now B is a minimal parabolic containing A, and P is any parabolic containing B.

In our main result we shall assume P self-opposed, but most of the argument does

not require this. Write the Levi decomposition as P = LN as before. We apply the recipe of ?5 to the restricted roots of A in G, and construct a reduced root

system A with positive roots A+, whose dual roots are individually proportional to the restricted roots. Let Ap, Ap be the roots in A, A+ whose duals are in L. Let

W and Wp be the Weyl groups of A in G and L. They are also the Weyl groups of the root systems A and Ap. Let wo and wp be the long words in W and Wp, and set vv =

wpwo.

Let N be the negative unipotent radical, that is, containing the negatives of

the roots in Af, and let P = LWNW denote the standard parabolic opposite to P. Let

S(p) = | det Ad(p)\u\ and 8(p) = | detAd(p)|#w| be the modular functions of P and P. Because it appears in the Plancherel measure, we shall compute the constant

7(G/P):= f 6i(n)-xdn, Jn

where 8\ is the extension of 8 to to G = KoP by left #o-invariance and dn is

the Haar measure on ^V assigning volume one io N C\Ko. This is the traditional

formula for ^(G/P), but we prefer the right /To-invariant extension 8r instead,

and it is easy to check that

7(G/P)= [ 6r(n)dn. JN

For a G A+, let Ca be as defined in ?5, and put

C = II ^ e QT).

a A+-A+

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADlC GROUPS 243

Proposition 11.1. Assume that P = P is self-opposed. Then

7(G/P) = C(4/2)>

where 8b is the modular function ofB.

If all qa = q, a well-known formula of Macdonald [Mac], applied both to

G and the Levi factor L, implies then that 7(G/P) =

P(q~x), where P(t) is the

Poincare polynomial of the flag manifold G/P, where " A"

denotes Langlands dual.

The first step in the proof of (11.1) is to view 8r as the unique ^-invariant

vector in the degenerate principal series representation Indp (82), taking value

one at the identity. As in ?7, we have an intertwining operator A: Indp (82) ?>

Ind? (W82) given by the analytic continuation of the integral

A(f)(g)= [f(w-xng)dn, Jn

with vol(_V n #0) = 1> and any representative vv of vv chosen in Ko. Now P being

self-opposed implies w2 = 1 and _V - wNw~x, so

7(G/P) = / 8r(w~xnw)dn = A(8r)(l). Jn

We shall compute A(8r)(l). The argument does not require P to be self-opposed, so we now drop that condition.

In case P = B is minimal, A(8r)(l) was computed in [Cas3] as the value of a

c-function. To generalize this to arbitrary parabolics, we might embed everything in full principal series, compare intertwining operators and invoke [Cas3]. The

problem is that the extension of A to the full principal series is no longer given by an integral, but the analytic continuation thereof. To compare, one must restrict to

Iwahori spherical functions, where problems of analytic continuation disappear, but that introduces further comparisons with Hecke module intertwiners. It is

therefore more efficient to ignore the full principal series, and go directly to

Hecke modules. What follows in analagous to Proposition 8.1, and if P = B,

gives a new proof of Casselman's result.

Let Ip(82) denote the X-invariants in Indp (?2). Let **W and Wp denote re

spectively the shortest coset representatives for Wp\W and W/Wp. Note that

w G ^W fl Wp. For x G ^VT let ipx G //>(<?) 2) be the unique function with support in Pxl such that ^x(x)

= 1. The i/jx form a basis of Ip(82), and 8r = ^2xePw^x

Let [Wp] be the characteristic function ch(P D Ko)T. In other words, [Wp] =

T,zewP Tz> where for any z G W, Tz = chlzl. For x G PW, put T'x

= [WP]TX =

244 MARK REEDER

I2zewP Tzx- Define

V: C (G)?>lndGP(8h V(f)(g) = jp8-\p)f(pg)dp,

(vol(P fl Ko) =

1). Letting tt denote the operation of the algebra C^?(G) on

Indp (?2), it is easy to verify the formulas

TT(<i))V(f) = V(f4>), V(T'x) = *px,

where (j)(g) =

4>(g~x), which together yield

Lemma 11.2. Fory G W andx G PW, we have

TT(Ty)xl>x =

V(TfxTy-i).

In particular, ipw generates the H-module Ip(8^).

The same is true for Ip(w8^) with the analogous definition of ipx, for x G PW.

As in ?5, the Hecke algebra H, now of J- bi-invariant functions, has the

decomposition H = H(K0)<8>C[T]. Let HP be the subalgebra generated by C[T] and the Tz for z G Wp. Let 8l be the modular function of B D L. There is

a unique algebra homomorphism r: Hp ? C defined by r(6)

= 6(6 2

8L 2),

t(Tz) = qz. Likewise, we have an algebra homomorphism wr: Hp

?> C defined i

by wt(0) = t(0w), wt(Tz) =

t(Twzw-\) = qz. As an element of T, r =

w/>(5j, where 8b is the modular function of B. In particular, r belongs to the open subset

To C T defined by the holomorphicity of all (a. We have W-modules

MP(r) = H?HpCr, Mp(WT)

= H ?Hp Cvr.

Set vT = 1 ? 1 G Mp(r), vwr = 1 <g) 1 G Mp(wr). Because r G To, the W-action on

Mp(r) extends to the algebra H(To), the localization as in ?8. Let vv = sk... sx

be a reduced expression with si = sQ. simple reflections, and as in ?8, consider

Fw = [TSk

- qak +

Cak] [^5, -

<?a, + Ca,] G W(T0).

Up to scalar, there is a unique H-module homomorphism F: Mp(vvr) ?

Mp(r)

determined by F(vWT) = FwvT. To see this, it suffices to check that FwvT trans

forms by wr under Hp. Let s = sa, with a e J, put F5 = Ts?qa+(a, and similarly

for 5 = W5VV- *. In H(To), we have the relation FSFW = Fsw = Fwj = F^Fj, from

which the claim follows by a simple computation.

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON /7-ADIC GROUPS 245

Lemma 11.3. [J, (2.1.2)] There are H-module isomorphisms

h: IP(8h ?

Mp(wT), h: Ip(w8h ?

Mp(t),

determined by h(ipw) = _Vr> MVv-O

= vr

By the uniqueness of F there is a scalar n making commutative the diagram

l kA I

lp(82) -

Ip(w82)

1 ?i F Mp(WT)

- Mp(t).

To determine k, we need more explicit formulas for h and h.

Lemma 11.4. There exist unique scalars cx,y G C, indexed by (x,y) G ̂xW^, such that

Tx =

_>__, ^Th,/^-!.

Moreover, cXy is nonzero only ify~x < wox in the Bruhat order, cXiW is nonzero only

ifx= 1 andcXyW =

q~x.

Proof. First let x G W be arbitrary. An easy induction on ?(x) shows that

TWox =

TWo ̂ 2y<xdXjyTy for some complex numbers dXyy. Therefore

Tx = TWo 2_^ dWoXyTy.

y<w0x

Assume now x G ̂ and left multiply by [Wp] to get

TX =

qwpTw 2^ "woxyTy y<w0x

For z G Wp, y G Wp, we have

TwTzy~] =

[WpI^wTz^-i =

[Wp]Twzw-\TwTy-\ =

qzT'wTy-\,

so we may take c^ =

<?wP Ylzew- qzdWoX,Zy-]> the z//l term of which is nonzero

only if zy~x < wqx. Since y~x < zy~x for all z, we have proved existence and

the first assertion in the "moreover." Since 1^1 =

\WP\, and {Tx: x G PW} are

linearly independent yet contained in the span of {T'wTy-\: y G Wp}, the latter

are linearly independent as well, proving uniqueness. For w = wpwo, the relation

w~~x < wox is equivalent to x < wp which implies x G Wp. But x G PW, so x = 1.

246 MARK REEDER

To prove the last assertion we define, for any v G W scalars Ryv by the

expansion

T~-\ = YRy^}Ch Ty

yew

Then Ryv is nonzero only if y < v, Rva, = 1 and Rsy,Sv

= Ry,v whenever sy > y,

sv > v. In particular,

(H-5) T~0l =

J2RUy~^yiTr yew

Left multiply both sides of (11.5) by [W] := J2z W Tz to get

^olW = E/?i..v-'w0'?;1^[^], V

hence

Y R^y =

(lwo yew

Applying this to Wp instead of W gives

(11.6) Yr^> =

^p' yeWp

Since vv0 = wPw, and ?(w0) =

?(wp) + ?(w), we have T~x = T~XT~X, so (11.5)

may be rewritten as

(H-7) T~p =TWY R\,yw0qy Ty-\. yew

Left multiply both sides of (11.7) by [WP] = YlzewP Tz> to get

qwXp[Wp\ =

TfwYRl,ywoqylTy-{ =

Y &yl Yl Rhyzw0]TwTy-^ yew yeWp zewp

as in the first part of the proof. It follows from uniqueness that

c\,w =

qwpqw / j *M,w2wo*

zewP

Now vvVYpWo = Wpvvvvo = Wpwp = Wp, so the latter sum is q~x by (11.6),

completing the proof of Lemma 11.4.

HECKE ALGEBRAS AND HARMONIC ANALYSIS ON p-ADIC GROUPS 247

Corollary 11.8. With cXy as in Lemma 11.4 and h as in Lemma 11.3, we have

h(i/jx) = Y CxjTyVwr Mp(wT).

yewP y-]<WQX

Of course a similar statement holds for h.

Now a computation entirely similar to one in ?8 shows that A(ipw) =

^l +

Y^x?\ ax^x- It then follows from Lemma 11.4 and Corollary 11.8 that the coef

ficient of TwvT in hA(ipw) is q~x. On the other hand, the coefficient of TwvT in

Fh(tpw) = FvWT is clearly one. We have proved

Lemma 11.9. F o h = qwA o h.

Now for the effect on spherical vectors. Let 8r be the analogue of 8r for P.

Let P(q) =

T,zeWp qz* and define

<\> = -^\W]vWT

= Y Tyv*T, 4> =

-^?AW]vT =

Y Tyvr nq) yew?

nq) yew?

From Corollary 11.8, we have

h(8r) =

q-x<f>, h(8r) = q-x$.

Now the relation [W]TS = qs[W] implies

FM = ^r-AW]F(vWT)

= ^f\[W]vT

= c(r)& P(q) P(q)

and it follows from the definition of 7(G/P) that

A(8r) = 7(G/P)?r.

By Lemma 11.9, we finally have

1(G/P)q~x4> = /U(<5r) =

q~XFh(8r) =

^2P(0) = <7"2C(r)0,

hence

7(G/P) =

<7-1C(r).

Since wp permutes the roots in A+ - Ap, we have ((t)

= ((wpt)

= C(<V ), and

the proof of Proposition 11.1 is complete.

248 mark reeder

Department of Mathematics, University of Oklahoma, Norman,

Oklahoma 73019

Electronic mail: mreeder@uoknor.edu

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