Definitions and preliminary results

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

Last update: 31 March 2013

Definitions and preliminary results

The Weyl group. Let R be a reduced irreducible root system in n, fix a set of positive roots R+ and let {α1,,αn} be the corresponding simple roots in R. Let W be the Weyl group corresponding to R. Let si denote the simple reflection in W corresponding to the simple root αi and recall that W can be presented by generators s1,s2,,sn and relations

si2 = 1, for1in, sisjsi mijfactors = sjsisj mijfactors , forij,

where mij= αi,αj αj,αi, and αi=2αi/ αi,αi.

The Iwahori-Hecke algebra. Fix q* such that q is not a root of unity. The Iwahori-Hecke algebra H is the associative algebra over defined by generators T1,T2,,Tn and relations

Ti2 = (q-q-1) Ti+1, for1in, TiTjTi mijfactors = TjTiTj mijfactors , forij, (1.1)

where mij are the same as in the presentation of W. For wW define Tw=Ti1Tip where si1sip=w is a reduced expression for w. By [Bou1968, Ch. IV §2 Ex. 23], the element Tw does not depend on the choice of the reduced expression. The algebra H has dimension W and the set {Tw}wW is a basis of H.

The group X. The fundamental weights are the elements ω1,,ωn of n given by ωi,αj =δij. The weight lattice is the W-invariant lattice in n given by


Let X be the abelian group P except written multiplicatively. In other words,

X={XλλP}, andXλXμ= Xλ+μ=XμXλ, forλ,μP.

Let [X] denote the group algebra of X. There is a W-action on X given by wXλ=Xwλ for wW, XλX, which we extend linearly to a W-action on [X].

The affine Hecke algebra. The affine Hecke algebra H associated to R and P is the algebra given by

H=-span { TwXλw W,XλX }

where the multiplication of the Tw is as in the Iwahori-Hecke algebra H, the multiplication of the Xλ is as in [X] and we impose the relation

XλTi=Ti Xsiλ+ (q-q-1) Xλ-Xsiλ 1-X-αi ,for1inand XλX. (1.2)

This formulation of the definition of H is due to Lusztig [Lus1989] following work of Bernstein and Zelevinsky. The elements TwXλ, wW, XλX, form a basis of H.

Weights. Let

T= { group homomorphismst:X* } .

The torus T is an abelian group with a W-action given by (wt)(Xλ) = t(Xw-1λ). For any element tT define the polar decomposition

t=trtc, trtcTsuch that tr(Xλ) >0,and tc(Xλ) =1,

for all XλX. Let Q=iαi. There is a unique μn and a unique νn/Q such that

tr(Xλ)= eμ,λ andtc (Xλ)= e2πiν,λ ,for allλP. (1.3)

In this way we identify the sets Tr={tTt=tr} and Tc={tTt=tc} with n and n/Q, respectively.

Central characters.

Theorem 1.4. (Bernstein, Zelevinsky, Lusztig [Lus1983, 8.1]) The center of H is [X]W= { f[X] wf=f } .

Since H has countable dimension, Dixmier’s version of Schur’s lemma implies that Z(H) acts on an irreducible H-module M by scalars. Let tT be such that

pM=t(p)M, for allpZ(H).

Since Z(H)=[X(T)]W it follows that t(p(X))=(wt)(p(X)) for all wW. The W-orbit Wt of t is the central character of M. We shall often abuse notation and refer to any weight sWt as “the central character” of M.

Weight spaces. Let M be a finite dimensional H-module. For each tT the t-weight space of M and the generalized t-weight space are the subspaces

Mt = { mMXλm =t(Xλ)m for allXλX } and Mtgen = { mMfor each XλX, (Xλ-t(Xλ))k m=0for somek >0 } ,

respectively. If Mtgen0 then Mt0. In general MtTMt, but we do have

M=tT Mtgen.

This is a decomposition of M into Jordan blocks for the action of [X]. The set of weights of M is the set

supp(M)= {tTMtgen0}. (1.5)

The calibration graph. Let tT. Define a graph Γ(t) with

Vertices: Wt, Edges: wtsiwt, if(wt) (Xαi) q±2.

Proposition 1.6. ([Ram1998] Proposition 2.12) Let M be a finite dimensional irreducible H-module with central character t. Then

dim(Msgen)= dim(Msgen)

if s and s are in the same connected component of the calibration graph Γ(t).

If tT define

P(t)= {α>0t(Xα)=q±2} andZ(t)= {α>0t(Xα)=1}. (1.7)

For each subset JP(t) define

(t,J)= { wWR(w) Z(t)=,R (w)P(t)=J } , (1.8)

where R(w)={α>0wα<0} is the inversion set of w. Define a placed shape to be a pair (t,J) such that tT, JP(t) and (t,J). The elements of the set (t,J) are called standard tableaux of shape (t,J).

Proposition 1.9. ([Ram1998] Theorem 2.14) Let tT. The connected components of the calibration graph Γ(t) are the sets

{wtw(t,J)}, JP(t), such that (t,J).

Calibrated representations. A finite dimensional H-module M is calibrated if Mtgen=Mt, for all tT.

Proposition 1.10. ([Ram1998] Proposition 4.2)

  1. An irreducible H-module M is calibrated if and only if dim(Mtgen)=1 for all weights t of M.
  2. If M is an irreducible H-module with regular central character t (i.e. Z(t)=) then M is calibrated.

Let αi and αj be simple roots in R and let Rij be the rank two root subsystem of R which is generated by αi and αj. Let Wij be the Weyl group of Rij, the subgroup of W generated by the simple reflections si and sj. A weight tT is calibratable for Rij if one of the following two conditions holds:

  1. t(Xα)1 for all αRij,
  2. Rij is of type C2 or G2 (assume that αi is the long root and αj is the short root), ut(Xαi)=q2 and ut(Xαj)=1 for some uWij, and t(Xαi)1 and t(Xαj)1.

A placed skew shape is a placed shape (t,J) such that for all w(t,J) and all pairs αi,αj of simple roots in R the weight wt is calibratable for Rij.

Theorem 1.11. ([Ram1998] Theorem 3.1 and Proposition 4.1)

  1. Let (t,J) be a placed skew shape and let (t,J) be the set of standard tableaux of shape (t,J). Define H(t,J)= -span {vww(t,J)} , so that the symbols vw are a labeled basis of the vector space H(t,J). Then the following formulas make H(t,J) into an irreducible H-module: For each w(t,J), Xλvw = (wt)(Xλ) vw, forXλX , and Tivw = (Ti)wwvw +(q-1+(Ti)ww) vsiw, for1in, where (Ti)ww= q-q-1 1-(wt)(X-αi) , and we set vsiw=0 if siw(t,J).
  2. If M is an irreducible calibrated representation such that supp(M)= {wtw(t,J)} for some placed skew shape (t,J) then M is isomorphic to the module H(t,J) constructed in (a).

Remark 1.12. It follows from the results of Rodier [Rod1981] that if M is an irreducible H-module with regular central character (i.e. Z(t)=) then M satisfies the hypothesis of the statement of Theorem 1.11 (b).

Langlands classification. The following discussion follows the work of Evens [Eve1996] and [KLu0862716, §8]. For this subsection it is convenient to assume that q>0 and q1. For the general case see [KLu0862716, §8]. Let tT and let t=trtc be the polar decomposition of t. Define

ν(t)i=1n αiby requiring tr(Xλ)= q2λ,ν(t) ,for allλP.

A finite dimensional H-module M is tempered if for all weights t of M (as defined in (1.5)) we have

ωi,ν(t) 0,for all1in.

The module M is square integrable if ωi,ν(t)<0 for all 1in. and all weights t of M.

Let I be a subset of the simple roots and let HI be the subalgebra of H generated by Ti, iI, and all XλX. We shall say that a finite dimensional HI-module is tempered if I is the maximal set such that for all weights t of M,

ωi,ν(t) 0,for alliI.

Theorem 1.13. (see [Eve1996]) Let I{1,2,,n} and let 𝒯 be an irreducible tempered representation of HI.

  1. M𝒯,I=IndHIH(𝒯) has a unique irreducible quotient L𝒯,I.
  2. Every irreducible H-module is isomorphic to L𝒯,I for some pair (𝒯,I).
  3. If L𝒯,IL𝒯,I then I=I and 𝒯𝒯 as HI-modules.

The Langlands parameters of an irreducible H-module M are given by the pair (𝒯,I) specified by Theorem 1.13 (b).

Classification by indexing triples. Kazhdan and Lusztig [KLu0862716] (see also the important work of Ginzburg [CGi1433132]) gave a refinement of the Langlands classification. Let G be the simple complex algebraic group with root system R and weight lattice P. An indexing triple (s,u,ρ) consists of

a semisimple elementsG, a unipotent elementuG, such thatsus-1 =uq2,

and an irreducible representation ρ of the component group A(s,u)= ZG(s,u)/ ZG(s,u), where ZG(s,u)= ZG(s) ZG(u). Let K(s,u) be the K-theory of the variety

s,u= { Borel subgroups ofGcontaining both sandu } .

By a theorem of Lusztig [Lus1985] K(s,u) is an H-module. The group A(s,u) also acts on K(s,u) and this action commutes with the action of H. The standard modules Ms,u,ρ are the H-modules given by the decomposition

K(s,u)= ρMs,u,ρ ρ,

where the sum is over all irreducible representations of A(s,u).

Theorem 1.14. [KLu0862716]

  1. If Ms,u,ρ0 then it has a unique simple quotient Ls,u,ρ.
  2. Every simple H-module isomorphic to some Ls,u,ρ.
  3. If Ls,u,ρ Ls,u,ρ then there is a gG such that s=gsg-1, u=gug-1, and ρ=ρ.

In this way each irreducible H-module corresponds to a unique (up to conjugation) indexing triple. One can replace u by n=lnu in the Lie algebra 𝔤=Lie(G) (see [CGi1433132, Ch. 8]) so that an indexing triple is

a semisimple elementsG, a nilpotent elementn𝔤, such thatAd(s)n= q2n,

and an irreducible representation ρ of the component group A(s,n)= ZG(s,n)/ ZG(s,n), where ZG(s,n)= ZG(s) ZG(n) and ZG(n) is taken with respect to the adjoint action of G on 𝔤. We will use this form of the indexing triples in the examples in later sections.

Principal series modules. Let tT and let vt be the one dimensional [X]-module corresponding to the character t:X*. Specifically, vt is the one dimensional vector space with basis {vt} and [X]-action given by

Xλvt=t(Xλ) vt,for allXλX.

The principal series module corresponding to t is the H-module

M(t)= IndXH (vt).

Theorem 1.15. [Mat1977]

  1. Every irreducible H-module M with central character t is a composition factor of the principal series module M(t).
  2. If wW and tT then M(t) and M(wt) have the same composition factors.

Theorem 1.16. Kato’s irreducibility criterion [Kat1981]) Let tT and let P(t)= { α>0t (Xα)=q±2 } . The principal series module M(t) is irreducible if and only if P(t)=.

Remark. Kato actually proves a more general result and thus needs a further condition for irreducibility. We have simplified matters by specifying the weight lattice P in our construction of the affine Hecke algebra. One can use any W-invariant lattice in n and Kato works in this more general situation. When the one uses the weight lattice P, a result of Steinberg [Ste1968-2, 4.2, 5.3] says that the stabilizer Wt of a point tT under the action of W is always a reflection group. Because of this Kato’s criterion takes a simpler form.

Weights of induced modules. If I{1,,n} define HI to be the subalgebra of H generated by Ti, iI, and all XλX.

Lemma 1.17. Let tT such that t(Xαi)=q2 for all iI and let vt be the one dimensional HI-module with basis {vt} and HI-action given by

Tivt=qvt, foriI,and Xλvt=t(Xλ) vt,for allXλ X.

Let W/WI be the set of minimal length coset representatives of cosets of WI in W. Then the weights of the H-module M=IndHIH(vt) are wt, wW/WI, and

dim(Mwtgen)= (# ofuW/WI such thatut=wt ).


The module M has basis {TwvtwW/WI}. By writing Tw=Ti1Tip for a reduced word w=si1sip and inductively using the defining relation (1.2) we get

Xλ(Twvt) = t(Xw-1λ) (Twvt)+ v<wav(t) (Tvvt) = (wt)(Xλ) (Twvt)+ v<wav(t) (Tvvt),

where the sum is over vW which are less than w in Bruhat order and av(t). This shows that the eigenvalues of Xλ on M are (wt)(Xλ). The result follows by counting the multiplicity of each eigenvalue.

The τ operators. The maps τi:MtgenMsitgen defined below are local operators on M in the sense that they act on each weight space Mtgen of M separately. The operator τi is only defined on weight spaces Mtgen such that t(Xαi)1.

Proposition 1.18. ([Ram1998] Proposition 2.7) Let tT such that t(Xαi)1 and let M be a finite dimensional H-module. Define

τi: Mtgen Msitgen m ( Ti- q-q-1 1-X-αi ) m.
  1. The map τi:Mtgen Msitgen is well defined.
  2. As operators on Mtgen, Xλτi= τiXsiλ, for all XλX.
  3. As operators on Mtgen, τiτi= (q-q-1Xαi) (q-q-1X-αi) (1-Xαi) (1-X-αi) .
  4. Let 1ijn and let mij be as in (1.1). Then τiτjτi mijfactors = τjτiτj mijfactors , whenever both sides are well defined operators on Mtgen.

Lemma 1.19. Let tT such that t(Xαi)=1 and suppose that M is an H-module such that Mtgen0. Let Wt be the stabilizer of t under the action of W on T. Assume that wW/Wt is such that t and wt are in the same connected component of Γ(t). Let w be a minimal length coset representative of w. Then

  1. dim(Mwtgen)2, and
  2. If Msjwtgen=0 then (wt)(Xαj) =q±2 and w-1αj,αi =0.


Let M(t) be the two dimensional principal series module for the affine Hecke algebra HA1 of type A1 (see §2 central character to). Then M(t)=M(t)tgen and has basis {vt,T1vt}. Let nt be a nonzero weight vector in Mt. There is a unique HA1-module homomorphism

M(t)M vtnt

where we view M as an HA1-module by restriction to the parabolic subalgebra H{i}H. This homomorphism must be an injection since M(t) is irreducible. Thus the vectors nt,Tint span a two dimensional subspace of Mtgen and XλX acts on this subspace by the matrix

ϕt(Xλ)=t (Xλ) ( 1(q-q-1)λ,αi 01 ) .

Let w=si1sip be a reduced expression of w. Since t and wt are in the same connected component of Γ(t) we can use Proposition 1.18 (c) to show that the map

τw=τi1 τip:Mtgen Mwtgen

is well defined and bijective. Thus the vectors τwnt, τwTint span a two dimensional subspace of Mwtgen and, by Proposition 1.18 (b) XλX acts on this subspace by the matrix wt

ϕwt(Xλ)=t (Xw-1λ) ( 1 (q-q-1) w-1λ,αi 01 ) .

This proves (a). Then

ϕwt (1-X-αj)= (1-t(X-w-1αj)) ( 1 (q-q-1)t (X-w-1αj) 1-t (X-w-1αj) -w-1,αj, αi 01 ) .

Since Msjwtgen=0, τj: Mwtgen Msjwtgen is the zero map and so

ϕwt(Tj)= ϕwt ( q-q-1 1-X-αj ) = q-q-1 1-t(X-w-1αj) ( 1 (q-q-1)t (X-w-1αj) 1-t (X-w-1αj) -w-1,αj, αi 01 ) .

The relation Tj2=(q-q-1)Tj+1 is the same as (Tj-q) (Tj+q-1) =0. This relation forces ϕwt(Tj) to have Jordan blocks of size 1 and eigenvalues ±q±1. It follows that t(Xw-1αj)=q±2 and w-1αj,αi=0.

Notes and References

This is an excerpt of a preprint entitled Representations of rank two affine Hecke Algebras, written by Arun Ram, Department of Mathematics, Princeton University, August 5, 1989.

Research supported in part by National Science Foundation grant DMS-9622985, and a Postdoctoral Fellowship at Mathematical Sciences Research Institute.

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