The affine Hecke algebra

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

Last update: 20 February 2013

The affine Hecke algebra

Though we shall never really use the data (G,B,T) it is conceptually useful to note that there is an affine Hecke algebra associated to each triple (GBT) where

An example of this data is when G=GLn(), B is the subgroup of upper triangular invertible matrices, and T is the subgroup of invertible diagonal matrices.

The reason that we can avoid the data (GBT) is that it is equivalent to different data (W,C,L) where

This will be our basic data. In the example where G=GLn() and B and T are the upper triangular and diagonal matrices, respectively,

W=Sn, 𝔥*=n= i=1nεi, C= { μ=i=1n μiεi μ1μn } ,L=i=1n εi, (1.1)

where W=Sn is the symmetric group, acting on 𝔥*=n by permuting the orthonormal basis ε1,,εn. This example will be treated in depth in Sections 5–7. We shall show that the labeling sets (t,J) for weight spaces of affine Hecke algebra representations that are introduced in (2.18) and Corollary 2.19 and used for the classification in Theorem 3.6 are generalizations of standard Young tableaux.

The components W and L in the data (W,C,L) are obtained from (GBT) by

W=N(T)/T,X= Hom(T,*)= {XλλL},

where N(T) is the normalizer of T in G and Hom(T,*) is the set of algebraic group homomorphisms from T to *. The notation is designed so that the multiplication in the group X is

XλXμ= Xλ+μ= XμXλ,for μ,λL, (1.2)

see [Bou1968, III Section 8]. The reflection (or defining) representation of the group W is given by its action on 𝔥*= Ln and with respect to a W-invariant inner product , on 𝔥* the group W is generated by reflections sα in the hyperplanes

Hα= { x𝔥* x,α=0 } ,αR+. (1.3)

See the picture which appears just before Theorem 1.17. The chambers are the connected components of 𝔥*- ( αR+ Hα ) and these are the fundamental regions for the action of W on 𝔥*. Fixing a choice of a fundamental chamber C corresponds to the choice of the set R+ of positive roots, which corresponds to the choice of B in G.

In our formulation we may view the set R+ as a labeling set for the reflecting hyperplanes Hα in 𝔥* and

C= { x𝔥* x,α>0 for allαR+ } . (1.4)

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.

For wW, the inversion set of W is

R(w)= { αR+ wαR- } , (1.5)

where R-=-R+. There is a bijection

W { fundamental chambers for Wacting on𝔥* } , W w-1C (1.6)

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).

The simple roots α1,,αn in R+ index the walls Hαi of the fundamental chamber C and the corresponding reflections s1,,sn generate W. In fact, W can be presented by generators s1,s2,,sn and relations

si2 = 1 for1in, sisjsi mijfactors = sjsisj mijfactors forij, (1.7)

where the (acute) angle π/mij between the hyperplanes Hαi and Hαj determines the value mij.

Fix q* with q2±1. The Iwahori–Hecke algebra H associated to (W,C) 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, (1.8)

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, Chapter IV, Section 2 Exercise 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 affine Hecke algebra H associated to (W,C,L) algebra given by

H=-span { TwXλ wW,XλX } (1.9)

where the multiplication of the Tw is as in the Iwahori–Hecke algebra H, the multiplication of the Xλ is as in (1.2) and we impose the relation

XλTi=Ti Xsiλ+ (q-q-1) Xλ-Xsiλ 1-X-αi ,for1in andXλX. (1.10)

This formulation of the definition of H is due to Lusztig [Lus1983] following work of Bernstein and Zelevinsky. The elements TwXλ, wW, XλX, form a basis of H.

The group algebra of X,

[X]=-span {XλλL}, (1.11)

is a subalgebra of H with a W-action obtained by linearly extending the W-action on X,

wXλ=Xwλ, forwW,XλX. (1.12)

Theorem 1.13 (Bernstein, Zelevinsky, Lusztig [Lus1983, 8.1]). The center of H is [X]W= { f[X] wf=ffor all wW } .



z=λL,wW cλ,wXλTw Z(H).

Let mW be maximal in Bruhat order subject to cγ,m0 for some γL. If m1 there exists a dominant μL such that cγ+μ-mμ,m=0 (otherwise cγ+μ-mμ,m0 for every dominant μL, which is impossible since z is a finite linear combination of XλTw). Since zZ(H) we have

z=X-μzXμ= λL,wW cλ,w Xλ-μTw Xμ.

Repeated use of the relation (1.10) yields

TwXμ= νL,vW dν,vXνTv

where dν,v are constants such that dwμ,w=1, dν,w=0 for νwμ, and dν,v=0 unless vw. So

z=λL,wW cλ,wXλTw= λL,wW νL,vW cλ,wdν,v Xλ-μ+νTv

and comparing the coefficients of XγTm gives cγ,m= cγ+μ-mμ,m dmμ,m. Since cγ+μ-mμ,m=0 it follows that cγ,m=0, which is a contradiction. Hence z=λLcλ Xλ[X].

The relation (1.10) gives

zTi=Tiz= (siz)Ti+ (q-q-1)z

where z[X]. Comparing coefficients of Xλ on both sides yields z=0. Hence zTi=(siz)Ti, and therefore z=siz for 1in. So z[X]W.

It is often convenient to assume that W acts irreducibly on 𝔥* and that the lattice L is the weight lattice

P= { x𝔥* x,α for allαR+ } =i=1n ωi, (1.14)

where the fundamental weights are the elements ω1,,ωn of n given by

ωi,αj =δij,where αi= 2αi αi,αi (1.15)

and δij is the Kronecker delta. Many facts are easier to state in this case and the general case can always be reduced to this one. We will make some further remarks on this reduction at the end of this section.

Consider the connected regions of the negative Shi arrangement 𝒜- [ALi1999,Shi1997,Shi1994,Shi1987,Sta1996,Sta1998], i.e., the arrangement of (affine) hyperplanes given by

Hα1+α2 Hα1 Hα2 Hα1+2α2 Hα1+α2-δ Hα1-δ Hα1-δ Hα2-δ Hα1+2α2-δ C s1C s2C s1s2C s2s1C s1s2s1C s2s1s2C λ1 λs1 λs2s1 λs2 λs1s2 λs1s2s1 λs2s1s2 λs1s2s1s2 The arrangement 𝒜- Fig. 1.

𝒜-= { Hα,Hα-δ αR+ } whereHα= { xn x,α=0 } ,Hα-δ= { xn x,α=-1 } . (1.16)

Each chamber w-1C, wW, contains a unique region of 𝒜- which is a cone, and the vertex of this cone is the point λw which appears in the following theorem.

Theorem 1.17 [Ste1975]. Suppose that W acts irreducibly on 𝔥* and that X={XλλP} where P is the weight lattice. The algebra [X] is a free [X]W-module with

basis {XλwwW}, whereλw= w-1 ( siw<w ωi ) .


The proof is accomplished by establishing three facts:

  1. Let fy,yW, be a family of elements of [X]. Then det(zfy) is divisible by αR+ (Xα-1) W/2 .
  2. det (zXλy) z,yW =α>0 (1-Xα) W/2 .
  3. If f[X] then there is a unique solution to the equation
wWaw Xλw=f, withaw [X]W.

(a) For each αR+ subtract row zfy from row sαzfy. Then this row is divisible by (1-X-α). Since there are W/2 pairs of rows (zfy,sαzfy) the whole determinant is divisible by (1-X-α) W/2 . For α,βR+ the factors (1-X-α) and (1-X-β) are coprime, and so det(zfy) is divisible by αR+ (1-X-α) W/2 . This product and the product in the statement of (a) differ by the unit (X2ρ) W/2 in [X].

(b) By (a), det(zXλy) is divisible by (Xα-1) W/2 . The top coefficient of det(zXλy) is equal to

zWz Xλz= zW isiz<z Xωi= i=1n X(W/2)ωi= (Xρ)W/2,

and the top coefficient of (Xα-1) W/2 is (X2ρ)W/2.

(c) Assume that ay[X]W are solutions of the equation yWXλy ay=f. Act on this equation by the elements of W to obtain the system of W equations

yW (zXλy)ay= zf,zW.

By (a) the matrix (zXλy)z,yW is invertible and so this system has a unique solution with ay[X]W. In fact, the ay can be obtained by Cramer’s rule. Cramer’s rule provides an expression for ay as a quotient of two determinants. By (a) and (b) the denominator divides the numerator to give an element of [X]. Since each determinant is an alternating function, the quotient is an element of [X]W.

Remark. In [Ste1975] Steinberg proves this type of result in full generality without the assumptions that W acts irreducibly on 𝔥* and L=P. Note also that the proof given above is sketchy, particularly in the aspect that the top coefficient of the determinant is what we have claimed it is. See [Ste1975] for a proper treatment of this point.

1.18. Deducing the HL representation theory from HP

It is often easier to work with the representation theory of H in the case when L=P. It is important to be able to convert from this case to the case of a general lattice L. If W acts irreducibly on 𝔥* then the lattice L satisfies

QLP,whereP= i=1ωi andQ=i=1αi

are the weight lattice and the root lattice, respectively. The group Ω=P/Q is a finite group (either cyclic or isomorphic to /2×/2). It corresponds to the center of the corresponding complex algebraic group. Let us denote the corresponding affine Hecke algebras by


according which lattice is used to make the group X.

Theorem 1.19 [RRa2003]. Then there is an action of the finite group P/L on HP, by ring automorphisms, such that

HL= (HP)P/L= { hHP gh=hfor allg P/L } ,

is the subalgebra of fixed points under the action of the group P/L.

This theorem is exactly what is needed to apply a (not very well known) version of Clifford theory to completely classify the representations of HL in terms of the representations of HP, see [RRa2003].

Notes and References

This is an excerpt of the paper entitled Affine Hecke algebras and generalized standard Young tableaux written by Arun Ram in 2002, published in the Academic Press Journal of Algebra 260 (2003) 367-415. The paper was dedicated to Robert Steinberg.

Research supported in part by the National Science Foundation (DMS-0097977) and the National Security Agency (MDA904-01-1-0032).

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