Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia

Last update: 25 April 2013


The graded Hecke algebra

Let W be a finite reflection group, defined by its action on its reflection representation 𝔥*. For each reflection sαW fix a root α in the -1 eigenspace of sα. The roots α are chosen so that the set R of roots is W-invariant. Then sα fixes a hyperplane

Hα= (+1eigenspace ofsα)= {x𝔥*|α(x)=0},

where we fix the linear function α𝔥=Hom(𝔥*,) so that α(α)=2. By fixing a nondegenerate symmetric W-invariant bilinear form on 𝔥* we may identify 𝔥 and 𝔥*, which will be used at times to view the Hα as lying in 𝔥. Then

sαx=x- x,αα ,for allx𝔥* ,where x,α= α(x). (2.1)

Fix simple roots α1,,αn in the root system for W and let si=sαi be the corresponding reflections.

By extension of scalars, W acts on the complexification 𝔥*=𝔥* and, in terms of its action on 𝔥*, W is a complex reflection group. Then W acts on the symmetric algebra S(𝔥*) which is naturally identified with the algebra of polynomial functions on the vector space 𝔥=Hom(𝔥*,).

Fix parameters cα, cα0, labeled by the roots, such that

cα=cwα, forwW.

If W acts irreducibly on 𝔥*, this amounts to the choice of one or two values, depending on whether there are one or two orbits of roots under the action of W. The group algebra of W is

W= -span{tw|wW} with multiplication twtw= tww.

The graded Hecke algebra is

=WS (𝔥*)

with multiplication determined by the multiplication in S(𝔥*) and the multiplication in W and the relations

xtsi=tsisi (x)+cαi x,αi, forx𝔥*, (2.2)

where α1, , αn𝔥 are the simple co-roots. More generally, it follows that for any pS(𝔥*),

ptsi=tsi si(p)+cαi Δi(p)and tsip=si (p)tsi+cαi Δi(p),

where Δi:S(𝔥*)S(𝔥*) is the BGG-operator given by

Δi(p)= p-si(p)αi forpS(𝔥*) .

Proposition 2.1 ([Lus1988, Theorem 6.5]). The center of the graded Hecke algebra is Z()=S(𝔥*)W, the ring of W-invariant polynomials on 𝔥.


If pS(𝔥*)W, then

ptsi=tsisi (p)+cαi p-si(p)αi =tsip+0= tsip,

and so p commutes with tsi. Therefore, S(𝔥*)WZ().

Let pZ() and write p=wWpwtw. Fix v of maximal length such that pv has maximal degree. Let μ𝔥* be regular, meaning that the stabilizer Wμ is trivial. Then

μp= wWμpw tw equalspμ= wWpwtwμ= wWpw ( (wμ)tw+ u<w cu,wμ tu ) ,

where cu,wμ. Comparing coefficients of tv yields

μpv=pv· (vμ).

So μ=(vμ) and thus v=1 since μ is regular. So pS(𝔥*). Comparing coefficients of tsi in

ptsi=si(p) tsi+cαi p-si(p)αi

shows that p=si(p) for all 1in. So pS(𝔥*)W. Thus Z()=S(𝔥*)W.

Harmonic polynomials

Let us briefly review the relationship between S(𝔥*), S(𝔥*)W, and harmonic polynomials [CGi1433132, §6.3]. Let x1,x2,,xn be an orthonormal basis of 𝔥 and define a symmetric bilinear form , on S(𝔥*) by

P,Q= (P()Q) |xi=0, forP,QS (𝔥*),

where P()= P ( x1 ,, xn ) and |xi=0 denotes specializing the variables to 0 (or, equivalently, taking the constant term). The monomials are an orthogonal basis of S(𝔥*),

x1λ1 xnλn, x1μ1 xnμn = ( (x1)λ1 (xn)λn x1μ1 xnμn ) |xi=0 = δλ1μ1 δλnμn λ1! λ2! λn!,

and so the bilinear form , is nondegenerate. The vector space of harmonic polynomials is the set of polynomials orthogonal to the ideal of S(𝔥*) generated by W-invariants in S(𝔥*) with constant term 0,

= ( fS(𝔥*)W |f(0)=0 ) ,andS(𝔥*) =S(𝔥*)W ,

as vector spaces. More precisely, if {hw} is a -basis of , then any fS(𝔥*) can be written uniquely in the form

f=wpwhw, pwS (𝔥*)W.

If the basis {hw} consists of homogeneous polynomials, then the number and the degrees of these polynomials are determined by the Poincaré polynomial of W,

PW(t)= k0dim (k)tk= i=1n 1-tdi1-t =wW t(w), (2.3)

where d1,,dn are the degrees of a set f1,,fn of homogeneous generators of S(𝔥*)W=[f1,,fn] and k is the kth homogeneous component of . In particular, dim()= Card({hw})= PW(1)=|W| and S(𝔥*) is a free module over S(𝔥*)W of rank |W|.

The following useful lemma is well known (see, for example, [Kat1982-2, Lemma 2.11]). Other related results can be found in [Ste1975] and [Hul1974].

Lemma 2.2 Let {bw}wW be a basis for the vector space of harmonic polynomials and let X be the |W|×|W| matrix given by

X= (z-1bw) z,wW .


detX=ξ· (α>0α) |W|/2 ,

where ξ is a nonzero constant in .


Note that if bw is another basis of and we write

bw=vW cvwbv, cvw,


X= (z-1bw) z,wW = (z-1bv) (cvw) and sodetX= ξdetX,

for some nonzero constant ξ=det((cvw)). Thus, by changing basis if necessary, we may assume that the bw are homogeneous.

Subtract row z-1bw from row sαz-1bw. Then this row is divisible by α. By doing this subtraction for each of the |W|/2 pairs {z-1,sαz-1} we conclude that det(X) is divisible by α|W|/2. Thus, since the roots are co-prime as elements of the polynomial ring S(𝔥*),

det(X)is divisible by (α>0α) |W|/2 .

The degree of α>0α|W|/2 is (|W|/2)Card(R+) and, using (2.3), the degree of det(X) is

deg(wWbw) = kkdim(k)= (ddtPW(t)) |t=1=wW (w) = wWCard(R(w)) =αR+ (|W|/2)= (|W|/2)Card(R+).

Since these two polynomials are homogeneous of the same degree, it follows that the quotient det(X)/(α0α)|W|/2 is a constant. If det(X)=0, then the columns of X are linearly dependent. In particular, there exist constants cw, not all zero, such that wcwbw=0. But this is a contradiction to the assumption that {bw} is a basis of . So det(X)0.

For each 1in let Δi*:S (𝔥*)S (𝔥*) be the operator which is adjoint to the BGG-operator Δi with respect to ,. A homogeneous basis {bw|wW} of the space of harmonic polynomials can be constructed by setting

bw=Δw*(1) ,whereΔw*= Δi1* Δi* for a reduced wordw=si1 si.

Weights and calibrated representations

The group W acts on

𝔥=Hom(𝔥*,) by(wγ)(x) =γ(w-1x),

for wW, γ𝔥 and x𝔥*.

The inversion set of an element wW is

R(w)= {α>0|wα<0}. (2.4)

The choice of the simple roots α1,,αn𝔥* determines a fundamental chamber

C= { λ𝔥| αi,λ >0,1in } (2.5)

for the action of W on 𝔥. For a root αR, the positive side of the hyperplane Hα is the side towards C, i.e. {λ𝔥|λ,α>0}, and the negative side of Hα is the side away from C. There is a bijection

W {fundamental chambers forWacting on𝔥} w w-1C (2.6)

and the chamber w-1C is the unique chamber which is on the positive side of Hα for αR(w) and on the negative side of Hα for αR(w).

If sα is a reflection in W which fixes γ𝔥, then γ,α=0. By [Ste1964, Theorem 1.5], [Bou1968, Ch. V §5 Ex. 8] the stabilizer Wγ of γ under the W-action is generated by the reflections which stabilize γ and so

Wγ= sα|αZ(γ) whereZ(γ)= {α|γ(α)=0}.

The orbit Wγ can be viewed in several different ways via the bijections

Wγ W/Wγ { wW|R(w) Z(γ)= } { chambers on the positive side ofHαfor αZ(γ) } , (2.7)

where the last bijection is the restriction of the map in equation (2.6). If γ is real and dominant (i.e. γ(α)0 for all αR), then Wγ is a parabolic subgroup of W and {wW|R(w)Z(γ)=} is the set of minimal length coset representatives of the cosets in W/Wγ.

Let M be a simple -module. Dixmier’s version of Schur’s lemma (see [Wal1988]) implies that Z() acts on M by scalars. Let γ𝔥 be such that

pm=γ(p)m,for all mM,pS (𝔥*)W.

The element γ is only determined up to the action of W since γ(p)=wγ(p) for all wW. Because of this, any element of the orbit Wγ is referred to as the central character of M.

Since = WS(𝔥*)= WS(𝔥*)W, the graded Hecke algebra is a free module over Z()=S(𝔥*)W of rank dim(W)dim()=|W|2. Since Z() acts on a simple -module by scalars, every simple -module is finite dimensional of dimension |W|2. Proposition 2.8(a) below will show that, in fact, the dimension of a simple -module is |W|.

Let M be a finite dimensional -module and let γ𝔥. The γ-weight space and the generalized γ-weight space of M are

Mγ = { mM|xm=γ (x)mfor allx 𝔥* } , (2.8) Mγgen = { mM|for allx 𝔥*, (x-γ(x))k m=0for somek >0 } . (2.9)


M=γ𝔥 Mγgen,

and we say that γ is a weight of M if Mγgen0. Note that Mγgen0 if and only if Mγ0. A finite dimensional -module

Miscalibrated ifMγgen= Mγfor allγ 𝔥. (2.10)

Tempered representations and the Langlands classification.

A weight λHom(𝔥*,) is determined by its values λ,αi on the simple roots. Define Re(λ) and Im(λ) in 𝔥=Hom(𝔥*,) by Re(λ),αi=Re(λ,αi) and Im(λ),αi=Im(λ,αi), and write

λ=Re(λ)+i Im(λ).

For any simple reflection sj, we have

sjλ=Re(λ)-Re (λ,αj) αj+iIm(λ)-i Im(λ,αj) αj=sjRe(λ) +isjIm(λ),

and so

R(wλ)=wRe(λ) ,for allwW.

Let ωi be the dual basis to αi in 𝔥 and let C be the closure of the fundamental chamber C𝔥 defined in (2.5). For λ𝔥 let λ0 be the point of C which is closest to Re(λ). This point is uniquely defined because of the convexity of the region C. Since λ0C and the ωi are on the boundary of C, there is a uniquely determined set I such that

λ0=jI cjωjwith cj>0,

and we say that the weight λ is I-tempered. For each I the set {ωj,αi|jI,iI} is a basis of 𝔥 and λ0 and I can, alternatively, be determined by the unique expansion

Re(λ)=jI cjωj+ iIdi αiwith cj>0anddi 0. (2.11)

Proposition 2.3 ((Lemma of Langlands) [Lan1989, Corollary 4.6], [Kna1986, Lemma 8.59]). Let λμ denote the dominance ordering on 𝔥. If λ,μ𝔥 such that λμ, then λ0μ0.

For any subset I{1,,n}, let I be the subalgebra of generated by tsi,iI, and all x𝔥*. An I-module M is tempered if all weights of M are I-tempered.

Theorem 2.4. Let L be a simple -module.

  1. There is a subset I{1,2,,n} and a simple tempered I-module U such that L is the unique simple quotient of IU. If I and I are subsets of {1,2,,n} and U and U are simple tempered I and I-modules, respectively, such that L is a quotient of both HIU and I, U, then I=I and UU as I-modules.


Let L be a simple -module. Let λ be a weight of L such that

λ0is a maximal element of {μ0|μis a weight ofL} (2.12)

with respect to the dominance ordering on 𝔥. Let I{1,2,,n} be determined by

λ0=jI cjωj

and let V be the I-submodule of L generated by a nonzero vector mλ in Lλ. Let WI be the subgroup of W generated by si, iI. The weights of V are of the form wλ with wWI. If wWI, then there are some constants αi such that

Re(wλ)= jIcj ωj+ αi0,iI aiαi+ αi>0,iI aiαi jIcj ωj+ αi0,iI aiαi,

since Re(λ) is as in (2.11). So, by Proposition 2.3,

(wλ)0 ( jI cjωj+ αi0 aiαi ) 0 =jI cjωj=λ0.

Thus, by the maximality of λ0,μ0=λ0 for all weights μ of V. So V is tempered.

Let U be a simple I-submodule of V. All weights of HI are of the form wμ with wW and μ a weight of U. Let WI denote the set of minimal length coset representatives of cosets in W/WI. If wμ is a weight and w=w1w2 with w1WI and w2WI, then by the argument just given w2μ is I-tempered and so

Re(w2μ)=jI cjωj+iI aiαiwith cj>0,ai0.

Recall that WI={w1W|R(w1){αi}iI=}. Thus, for each iI, w1αi is a positive co-root and

Re(w1w2μ)= w1(w2μ)0+ iIaiw1 αiw1 (w2μ)0.

If w11, then w1ωjωj for all jI and w1ωj<ωj for some jI. So

Re(w1w2μ) w1(w2μ)0 <(w2μ)0

and thus, by Proposition 2.3,

(w1w2μ)0 <(w2μ)0 whenw11. (2.13)

Let ν be a weight of U such that, among weights of U, ν0 is maximal. If N is an -submodule of HIU such that Nν0, then, by (2.13), NνUν and so NU0. Since U is simple as an I-module, any vector of U generates all of HIU and so N=HIU. This shows that if

Mmax= ( sum of all-submodulesN ofHIU such thatNν=0 ) ,

then Mmax is equal to the sum of all proper submodules of HIU and is the (unique) maximal proper submodule of HIU. So HIU has a unique simple quotient.

Since U is an I-submodule of L and induction is the adjoint functor to restriction, there is an -module homomorphism

ϕU: HIU L u u foruU. (2.14)

Thus, since L is simple, L(HIU)/Mmax. This proves (a) and shows that for any tempered I-module U the module HIU has a unique simple quotient.

To prove (b) let us analyze the freedom of the choices that are made in the above construction of HIU. Equation (2.13) and Proposition 2.3 show that ν0λ0 for all weights ν of HIU. In particular, all weights ν of L satisfy ν0λ0 and so λ0 is the same for all weights λ of L which satisfy (2.12). This shows that there is a unique choice of I in the construction of HIU. If U is another simple I-submodule of V, then either UU=0 or U=U. Suppose that UU=0. Then UU is a tempered I-submodule of L. Let ν be a weight of U. Suppose μ is a weight of L with μ0=ν0. By equations (2.13) and (2.14), the only elements of the μ-weight space of the image of the homomorphism ϕU:HIUL are elements of U. Thus im(ϕU)U=0. But this is impossible because L is simple and ϕU is surjective. Thus U=U.

Theorem 2.4 gives us a way to classify simple H-modules. The Langlands parameters (U,I) of the simple module L are the pair determined by Theorem 2.4.

τ operators

The following proposition defines maps τi:MγgenMsiγgen on generalized weight spaces of finite-dimensional -modules M. These are “local operators” and are only defined on weight spaces Mγgen such that γ(αi)0. In general, τi does not extend to an operator on all of M.

Proposition 2.5. Let M be a finite dimensional -module. Fix i, let γ𝔥 be such that γ(αi)0 and define

τi: Mγgen Msiγgen m (tsi-cαiαi)m.
  1. The map τi:MγgenMsiγgen is well defined.
  2. As operators on Mγgen, xτi=τisi(x) for all xS(𝔥*).
  3. As operators on Mγgen, τiτi= (cαi+αi) (cαi-αi) (αi)(-αi) .
  4. Both maps τi:MγgenMsiγgen and τi:MsiγgenMγgen are invertible if and only if γ(αi)±cαi.
  5. If 1i,jn,ij, let mij be the order of sisj in W. Then τiτjτi mijfactors = τjτiτj mijfactors , whenever both sides are well defined operators on Mγgen.


Since αi acts on Mγgen by γ(αi) times a unipotent transformation, the operator αi on Mγgen has nonzero determinant and is invertible. Since cαi/αi is not an element of S(𝔥*) or it will be viewed only as an operator on Mγgen in the following calculations.

If x𝔥* and mMγgen, then

xτim = x(tsi-cαiαi) m = ( tsisi(x)+ cαix,αi -cαixαi ) m = ( tsisi(x)- cαi x-x,αiαi αi ) m = ( tsisi(x)- cαisi(x)αi ) m = (tsi-cαiαi) si(x)m = τisi(x)m.

This proves (a) and (b).

τiτim = ( tsi2- cαiαi tsi-tsi cαiαi +cαi2αi2 ) m = ( 1-cαiαi tsi- cαi-αi tsi-cαi ( cαiαi- cαi-αi ) αi +cαi2αi2 ) m = ( 1+ cαi2 (αi) (-αi) ) m= ( (cαi+αi) (cαi-αi) (αi)(-αi) ) m,

proving (c).

(d) Since αi acts on Mγgen by γ(αi) times a unipotent transformation, det((cαi+αi)(cαi-αi))=0 if and only if γ(αi)=±cαi. Thus τiτi, and each factor in this composition, is invertible if and only if γ(αi)±cαi.

(e) Let w=si1si be a reduced word for wW and set τw=τi1τi. Using the definition τi=tsi-cαiαi and the defining relation (2.2) for yields an expansion

τw=tw+z<w Rztz,

where the Rz are rational functions of αR. We shall show that this expansion of τw does not depend on the choice of reduced word of w.

Let 1 be the trivial W-module and let e=wWtw. View the -module e IndW(1) =W1= S(𝔥*)WW1 =S(𝔥*)1 simply as S(𝔥*). Let us first show that this -module S(𝔥*)=IndW(1) is faithful. Assume that h=zWPztz in =S(𝔥*)W satisfies h(p)=0 for all pS(𝔥*). We must show that h=0.

Since 0=h(1)=zPz, and this is true degree by degree, we may assume that the polynomials Pz are homogeneous of the same degree. Use the notations of Lemma 2.2 so that {bw|wW} is a basis of the space of harmonic polynomials consisting of homogeneous polynomials. Then, for each wW,

0=h(bw)= zWPz tzbw(1) = ( zWPz (z-1bw) tz+lower degree terms ) (1) = zWPz (z-1bw)+ lower degree terms.

where, by definition, each tz is degree 0. Focusing on top degree terms in this equality, 0=zWPz(z-1bw), for each wW. By Lemma 2.2, the matrix (z-1bw)z,wW is invertible, so there is a nonzero ξ with ξ·(α>0α)|W|/2Pz=0, for every zW. Since S(𝔥*) is an integral domain, Pz=0 for each zW, and hence h=0. So the -module IndW(1)S(𝔥*) is faithful.

Let τi=tsiαi-cαi. As operators on IndW(1)S(𝔥*),

τiαi=τi, τi(1)= (-αitsi+cαi) (1)=(-αi+cαi) ,andτip= (sip)τi,

for any polynomial pS(𝔥*). Using the fact [Bou1968, Chapt. VI §1.11 Prop. 33] that, for a reduced word w=si1si, R(w)={αi,siαi-1,,sisi2αi1},

(τi1τi) (αR(w)α) (p) = ( ti1 ti ) p(1)=(wp) ( ti1 ti ) (1) = (wp) ( αR(w) (-α+cα) ) .

Thus, since S(𝔥*) is an integral domain and IndW(1) is faithful, τi1τi does not depend on the choice of the reduced word w=si1si.

Let γ𝔥 and define

Z(γ)= {α>0|γ(α)=0} andP(γ)= {α>0|γ(α)=±cα}. (2.15)

If JP(γ), define

(γ,J)= { wW| R(w)Z(γ) =andR(w) P(γ)=J } . (2.16)

A local region is a pair (γ,J) such that γ𝔥, JP(γ), and (γ,J). Under the bijection (2.6) the set (γ,J) maps to the set of points x𝔥 which are

  1. on the positive side of the hyperplanes Hα for αZ(γ),
  2. on the positive side of the hyperplanes Hα for αP(γ)\J, and
  3. on the negative side of the hyperplanes Hα for αJ.

In this way the local region (γ,J) really does correspond to a region in 𝔥. This is a connected convex region in 𝔥 since it is cut out by half spaces in 𝔥n. The elements w(γ,J) index the chambers w-1C in the local region and the sets (γ,J) form a partition of the set {wW|R(w)Z(γ)=} (which, by (2.7), indexes the cosets in W/Wγ).

Corollary 2.6. Let M be a finite dimensional -module. Let γ𝔥 and let JP(γ). Then

dim(Mwγgen)= dim(Mwγgen) forw,w (γ,J),

where (γ,J) is given by (2.16).


Suppose w,siw(γ,J). We may assume siw>w. Then α=w-1αi>0, αR(w) and αR(siw). Now, R(w)P(γ)=R(siw)P(γ) implies γ(α)±cαi. Since cα=cwα=cαi, wγ(αi)=γ(w-1αi)=γ(α)0 and wγ(αi)±cαi and thus, by Proposition 2.5(d), the map τi:MwγgenMsiwγgen is well defined and invertible. It remains to note that if w,w(γ,J), then w=si1siw where siksiw(γ,J) for all 1k. This follows from the fact that (γ,J) corresponds to a connected convex region in 𝔥.

The following lemma will be used in the classification in Section 3 to analyze weight spaces for representations with nonregular central character.

Lemma 2.7. Let γ𝔥 such that γ(αi)=0. Let M be an -module such that Mγgen0 and let w(γ,). Then

  1. dimMwγgen2,
  2. if Msjwγgen=0, then (wγ)(αj)=±cαj and w-1αj,αi=0.


Let A1 be the subalgebra of generated by tsi and all xS(𝔥*). Let vγ be the one-dimensional representation of S(𝔥*) defined by xvγ=γ(x)vγ and let M(γ)= IndS(𝔥*)A1 (vγ) = A1S(𝔥*) vγ. This module is irreducible and has basis {vγ,tsivγ} and, with respect to this basis, the action of x𝔥* on M(γ) is given by the matrix

ργ(x)= ( γ(x) cαix,αi 0 γ(x) ) . (2.17)

Let nγ be a nonzero vector in Mγ. As an S(𝔥*)-module nγvγ and, since induction is the adjoint functor to restriction, there is a unique A1-module homomorphism given by

M(γ)M vγnγ

Since M(γ) is irreducible, this homomorphism is injective, and the vectors nγ,tsinγ span a two-dimensional subspace of Mγgen on which the action of x𝔥* is given by the matrix in (2.17).

Let w=si1sip be a reduced word for w. Proposition 2.5(d) and the assumption that w(γ,) guarantee that the map

τw=τi1 τi: Mγgen Mwγgen

is well-defined and bijective. Thus τwnγ and τwtsinγ span a two-dimensional subspace of Mwγgen and, by Proposition 2.5(b), the A1 action of xX on this subspace is given by

ρwγ(x)= ( γ(w-1x) cαiw-1x,αi 0 γ(w-1x) ) .

This proves (a).

Assume Msjwγgen=0. Then part (a) implies sjwγwγ, so (wγ)(αj)=γ(w-1αj)0. So the matrix ρwγ(αj) is invertible and

ρwγ (1αj)= 1γ(w-1αj)2 ( γ(w-1αj) -cαiw-1αj,αi 0 γ(w-1αj) ) .

Since Msjwγgen=0, the map τj:MwγgenMsjwγgen is the zero map and

ρwγ(tsj) =ρwγ (cαjαj)= cαjγ(w-1αj)2 ( γ(w-1αj) -cαjw-1αj,αi 0 γ(w-1αj) ) .

Since tsj2-1=(tsj-1)(tsj+1)=0,ρwγ(tsj) must have Jordan blocks of size 1 and eigenvalues ±1. Since cαi0, it follows that γ(w-1αj)=±cαj and w-1αj,αi=0.

Principal series modules

For γ𝔥 let vγ be the one-dimensional S(𝔥*)-module given by

xvγ=γ(x) vγforx 𝔥*.

The principal series representation M(γ) is the -module defined by

M(γ)= S(𝔥*) vγ= IndS(𝔥*) (vγ). (2.18)

The module M(γ) has basis {twvγ|wW} with W acting by left multiplication. By the defining relations for , for x𝔥*, wW,

xtwvγ= (wγ)(x) twvγ+ z<wczw (x)tzvγ withczw (x).

Thus, if γ𝔥 is regular all the wγ are distinct and

M(γ)=wW m(γ)wγwith dim(M(γ)wγ) =1.

Thus, if γ𝔥 is regular, there is a unique basis {vwγ|wW} of M(γ) determined by

xvwγ=(wγ) (x)vwγ for allwWandx 𝔥*, (2.19) vwγ=tw vγ+u<w awu(γ) (tuvγ) whereawu(γ) . (2.20)


vwγ=τwvγ (2.21)

where τw=τi1τi2τip for a reduced word w=si1sip of w. The uniqueness of the element vwγ given by the conditions (2.19) and (2.20) shows that vwγ=τwvγ does not depend on the reduced decomposition which is chosen for w.

Part (a) of the following proposition implies that the dimension of every irreducible -module is less than or equal to |W|. In combination, part (a) and part (b) show that every irreducible -module with regular central character is calibrated. Part (c) is a graded Hecke analogue of a result of Rogawski [Rog1985, Proposition 2.3].

Proposition 2.8.

  1. If M is an irreducible finite dimensional -module with Mγgen0, then M is a quotient of M(γ).
  2. If γ𝔥 is regular, then M(γ) is calibrated.
  3. For fixed γ𝔥 and any wW, M(γ) and M(wγ) have the same composition factors.


(a) Since S(𝔥*) is commutative, an irreducible S(𝔥*) submodule must be one-dimensional. Thus there exists a nonzero vector mγ in Mγ and, as an S(𝔥*)-module, mγvγ. Since induction is the adjoint functor to restriction there is a unique -module homomorphism given by

M(γ) M vγ mγ

and, since M is irreducible, this homomorphism is surjective. Thus M is a quotient of M(γ).

(b) Since γ is regular, Wγ={1}, and by equation (2.19),

M(γ)=wW M(γ)wγand dim(M(γ)wγ) =1

for all wW. Since M(γ)wγ is nonzero whenever M(γ)wγgen is nonzero and dim(M(γ)wγgen)=1, M(γ)wγ=M(γ)wγgen for all wW.

(c) Let si be a simple reflection such that siγγ. Then γ(αi)0 and the operator τ is well defined on M(siγ)siγgen. The vector vsiγ is a weight vector in M(siγ)siγ and, by Proposition 2.5(b), τivsiγ is a weight vector of weight γ (it is nonzero since tsivsiγ and (siγ)(cαi/αi)vsiγ are linearly independent in M(siγ)). Thus, there is an -module homomorphism

A(si,γ): M(γ) M(siγ) hvγ hτivsiγ, h.

The modules M(γ) and M(siγ) have bases

{ tw (tsi+1) vγ, tw (tsi-1) vγ } siw>w , { tw (tsi+1) vsiγ, tw (tsi-1) vsiγ } siw>w , (2.22)

respectively. Since (tsi+1)tsi=tsi+1 and (tsi-1)tsi=-(tsi-1),

A(si,γ) (tw(tsi+1)vγ) = tw(tsi+1) ( tsi- cαiαi ) vsiγ = tw(tsi+1) (1-cαiαi) vsiγ = ( siγ ( αi-cαi αi ) ) tw(tsi+1) vsiγ A(si,γ) (tw(tsi-1)vγ) = tw(tsi-1) ( tsi- cαiαi ) vsiγ = tw(tsi-1) ( -1- cαiαi ) vsiγ = ( siγ ( αi+cαi -αi ) ) tw(tsi-1) vsiγ

and so the matrix of A(si,γ) with respect to the bases in (2.22) is diagonal with |W|/2 diagonal entries equal to (siγ)((αi-cαi)/αi) and |W|/2 diagonal entries equal to (siγ)((αi+cαi)/(-αi)). If γ(αi)±cαi, then A(si,γ) is an isomorphism and so M(γ) and M(siγ) have the same composition factors. If γ(αi)=±cαi, then dim(kerA(si,γ))=|W|/2. In this case A(si,siγ)A(si,γ)=0 and so the sequence

M(γ) A(si,γ) M(siγ) A(si,siγ) M(γ)

is exact. Then

M(siγ) ker(A(si,siγ)) 0

is a filtration of M(siγ) where the first factor is isomorphic to a submodule of M(γ),

M(siγ)/ ker(A(si,siγ)) im(A(si,siγ)) M(γ),

and the second factor is isomorphic to a quotient of M(γ),

ker(A(si,siγ)) M(γ)/ker(A(si,γ)).

Since dim(ker(A(si,siγ)))+ dim(im(A(si,siγ))) =|W|/2+ |W|/2= dim(M(siγ))= dim(M(γ)), it follows that M(γ) and M(siγ) must have the same composition factors.

Our next goal is to prove Theorem 2.10 which determines exactly when the principal series module M(γ) is irreducible. Let γ𝔥 and let M(γ)=S(𝔥*)vγ be the corresponding principal series module for . The spherical vector in M(γ) is

1γ= wWtw vγ. (2.23)

Up to multiplication by constants this is the unique vector in M(γ) such that tw1γ=1γ for all wW. The following proposition provides a graded Hecke analogue of the results in [Kat1982-2, Proposition 1.20] and [Kat1982-2, Lemma 2.3]. Mention of this analogue was made in [Opd1995].

Proposition 2.9.

  1. If γ is a generic element of 𝔥 and vwγ, wW, is the basis of M(γ) defined in (2.21), then 1γ=zW γ(cz)vzγ wherecz= αR(w0z) α+cαα.
  2. The spherical vector 1γ generates M(γ) if and only if α>0(γ(α)+cα)0.
  3. For γ𝔥, the principal series module M(γ) is irreducible if and only if 1wγ generates M(wγ) for all wW.


(a) Suppose that ξz are constants such that

1γ= (wWtw) vγ=zW ξzvzγ.

We shall prove that the ξ are given by the formula in the statement of the proposition. Since tsi (wWtw) = wWtw,

1γ = tsi1γ= ( τi+ cαiαi ) zW ξzvzγ= ( τi+ cαiαi ) siz>z ( ξzvzγ+ ξsiz vsizγ ) = siz>z ( ξzvsizγ+ ξzcαiγ(z-1αi) vzγ+ξsiz τi2vzγ+ ξsiz cαiγ(-z-1αi) vsizγ ) .

Comparing coefficients of vsizγ on each side of this expression gives

ξsiz=ξz+ ξsiz cαi γ(-z-1αi) ,

and so

ξzξsiz=γ ( z-1αi+cαi z-1αi ) ,ifsiz>z.

Using this formula inductively gives

ξw = ξsi1sip =γ ( sipsi2αi1 sipsi2αi1+cαi ) γ ( αip αip+cαip ) ξ1 = γ ( αR(w) αα+cα ) ξ1.

Since the transition matrix between the basis {twvγ} and the basis {vwγ} is upper unitriangular with respect to Bruhat order, ξw0=1. Thus, the last equation implies that

ξ1=γ ( α>0 α+cαα )


ξw=γ ( αR(w) αα+cα ) ·ξ1=γ ( αR(w0w) α+cαα ) .

(b) By expanding vzγ=τzvγ=τi1τipvγ for a reduced word si1sip=z it follows that there exist rational functions muz such that

vzγ=uW γ(muz)tu vγ,

for all generic γ𝔥. Furthermore, the matrix M=(muz)u,zW with these rational functions as entries is upper unitriangular.

Let bw, wW, be a basis of harmonic polynomials and define polynomials quyS(𝔥*), u,yW, by

by(wWtw) =uWtu quy,yW,

where these equations are equalities in . Then,

by1γ=by (wWtw) =uWγ (quy) (tuvγ),

and part (a) implies that if γ is generic, then

by1γ = byzWγ (cz)vzγ =zWγ (cz(z-1by)) vzγ = z,uWγ (cz(z-1by)muz) (tuvγ).

Since these two expressions are equal for all generic γ𝔥, it follows that

quy=zW muz·cz· (z-1by), u,yW, (2.24)

as rational functions (in fact, both sides are polynomials).

Since tw, wW, and pZ()=S(𝔥*)W act on 1γ by constants, the -module M(γ) is generated by 1γ if and only if there exist constants pyw such that

twvγ=yW pywby1γ for eachwW.

If these constants exist, then, for each wW,

twvγ= yWpyw by1γ= y,z,uWγ ( muzcz (z-1by) pyw ) tuvγ,

where, by (2.24), there is no restriction that γ be generic. If

M=(muz)u,zW, C=diag(cz)zW, X=(z-1by)z,yW, P=(pyw)y,wW,

then P=(γ(MCX))-1 and so P exists if and only if det(γ(MCX))0. Now det(M)=1, and, by Lemma 2.2 and part (a),

det(X)=ξ· α>0 α|W|/2


det(C)=zW αR(w0z) α+cαα= ( α>0 α+cαα ) |W|/2 ,

where ξ is nonzero. Thus P exists if and only if α>0(γ(α)+cα)0.

(c) : If M(γ) is irreducible, then, by Proposition 2.8(c), M(wγ) is irreducible for all wW. Hence M(wγ) is generated by 1wγ.
: Suppose that 1wγ generates M(wγ) for all wW. Let E be a nonzero irreducible submodule of M(γ) and let wW be such that the weight space Ewγ is nonzero. Then, by Proposition 2.8(a), there is a nonzero surjective -module homomorphism φ:M(wγ)E. Since 1wγ generates M(wγ), φ(1wγ) is a nonzero vector in E such that tvφ(1wγ)=φ(1wγ) for all vW. Since there is a unique, up to constant multiples, spherical vector in M(γ), ϕ(1wγ) is a multiple of 1γ and 1γ is nonzero. This implies that E=M(γ)since 1γ generates M(γ).

Together the three parts of Proposition 2.9 prove the following graded Hecke algebra analogue of [Kat1982-2, Theorem 2.1].

Theorem 2.10. Let γ𝔥 and let P(γ)={α>0|γ(α)=±cα}. The principal series -module

M(γ)is irreducible if and only if P(γ)=.

Notes and References

This is an excerpt of a paper entitled Representations of graded Hecke algebras, written by Cathy Kriloff (Department of Mathematics, Idaho State University, Pocatello, Idaho 83209-8085) and Arun Ram.

Research of the first author supported in part by an NSF-AWM Mentoring Travel Grant. Research of the second author supported in part by National Security Agency grant MDA904-01-1-0032 and EPSRC Grant GR K99015.

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