Affine Hecke algebras of general type

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

Last update: 28 February 2013

Affine Hecke algebras of general type

Let R be a reduced root system and let R be the root system formed by the coroots α=2α/α,α for αR. Let W be the Weyl group of R and fix a system of positive roots R+ in R. Let {α1,,αn} be the corresponding simple roots and let s1,,sn be the corresponding simple reflections in W. The fundamental weights are defined by the equations ωi,αj=δij and the lattices

P=i=1nωi andQ=i=1n αi,

are the weight lattice and the root lattice, respectively. The Dynkin diagrams and the corresponding extended Dynkin diagrams are given in Figure 3. If Γ is a Dynkin diagram or extended Dynkin diagram define

mij= { 2, i j if , 3, i j if , and mij= { 4, i j if , 6, i j if .

4.1 Affine Weyl groups

The extended affine Weyl group is the group

W=WP= { wtλ wW,λP } ,wherewtλ= twλw,

for wW and λP where tλ corresponds to translation by λP. Define s0W by the equation

s0sϕ= tϕ,where ϕis the highest root of R, (4.2)

see [Bou1968 Ch. IV §1 no. 2.1]. The subgroup Waff=WQ of W is presented by generators s0,s1,,sn and relations

si2=1,0in ,and sisj mij = sjsi mij ,ij,

where the mij are determined from the extended Dynkin diagram of the root system R. Define

Ω= {giωiis minuscule} ,wheregiw0 w0,i=tωi, (4.3)

w0 is the longest element of W and w0,i is the longest element of the group sj1j n,ji , see [Bou1968 Ch. IV §2 Prop. 6]. Then ΩP/Q and each element gΩ corresponds to an automorphism of the extended Dynkin diagram of R, in the sense that

if gΩthen gsig-1= sσ(i), (4.4)

where σ is the permutation of the nodes determined by the automorphism. Equation (4.4) means that W=WaffΩ. The usual length function on the Coxeter group Wff is extended to the group W by

(wg)=(w), forwWaffand gΩ.

Let L be a lattice such that QLP. View L/Q as a subgroup of ΩP/Q and let

WL=WL=Waff (L/Q).

Then Waff=WQ, W=WP, and WL is a subgroup of W.

4.5 Affine braid groups

Let L be a lattice such that QLP. The affine braid group L is the group given by generators Tw, wWL, and relations

TwTw= Tww,if (ww)=(w) +(w).

Let aff=Q. View L/Q as a subgroup of ΩP/Q. Then L=affL/Q is presented by generators Ti=Tsi, 0in, and relations

TiTj mij = TjTi mij ,andgTi g-1=Tσ(i) ,forgΩ, (4.6)

where σ is as in (4.4), and the mij are specified by the extended Dynkin diagram of R.

Let P+=i=1n0ωi be the dominant weights in P. Define elements Xλ, λP by

Xλ = Ttλ,if λP+,and Xλ = Xμ (Xν)-1, ifλ=μ-ν withμ,ν P+. (4.7)

By [Mac1423624, 3.4] and [Lus1989], the Xλ are well defined and do not depend on the choice λ=μ-ν, and

XλXμ=XμXλ =Xλ+μ,for λ,μP. (4.8)

Then XλL if and only if λL.

4.9 Affine Hecke algebras

Fix q*. The affine Hecke algebra HL is the quotient of the group algebra L by the relations

(Ti-q) (Ti+q-1) =0,for0i n. (4.10)

In HL (see [Mac1995, 4.2]),

XλTi=Ti Xsiλ+ (q-q-1) Xλ-Xsiλ 1-X-αi ,forλL, 1in. (4.11)

The Iwahori-Hecke algebra H is the subalgebra of H generated by T1,,Tn.

To our knowledge, the following theorem is due to Bernstein and Zelevinsky in type A, and to Bernstein in general type (unpublished). Lusztig has given an exposition in [Lus1989]. We give a new proof which we believe is more elementary and more direct.

Theorem 4.12. (Bernstein, Zelevinsky, Lusztig [Lus1989]) Let L be a lattice such that QLP, where Q is the root lattice and P is the weight lattice of the root system R. Let H=HL be the affine Hecke algebra corresponding to L and let [X]=span {XλλL}. Let W be the Weyl group of R. Then the center of H is

Z(H)= [X]W= { f[X]w f=ffor everywW } .



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-μTwXμ .

Repeated use of the relation (4.11) 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,w dν,v Xλ-μ+νTv

and comparing the coefficients of XγTw 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=λL cλXλ [X].

The relation (4.11) 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.

4.13 Deducing the HL representation theory from HP

Although we have not taken this point of view in our presentation, the affine Hecke algebras defined above are naturally associated to a reductive algebraic group G over [KLu0862716] or a p-adic Chevalley group [IMa1965]. In this formulation, the lattice L is determined by the group of characters of the maximal torus of G. It is often convenient to work only with the adjoint version or only with the simply connected version of the group G and therefore it seems desirable to be able to derive the representation theory of the affine Hecke algebras HL from the representation theory of the affine Hecke algebra HP. The following theorem shows that this can be done in a simple way by using the extension of Clifford theory in the Appendix. In particular, Theorem A.13, can be used to construct all of the simple HL-modules from the simple HP-modules.

Theorem 4.14. Let L be a lattice such that QLP, where Q is the root lattice and P is the weight lattice of the root system R. Let H=HL be the affine Hecke algebra corresponding to L. Then there is an action of a finite group K on HP, acting by automorphisms, such that

HL= (HP)K,

is the subalgebra of fixed points under the action of the group K.


There are two cases to consider, depending on whether the group ΩP/Q is cyclic or not.

Type An-1 Bn Cn D2n-1 D2n E6 E7 E8 F4 G2 Ω /n /2 /2 /4 /2×/2 /3 /4 1 1 1

In each case we construct the group L and its action on HP explicitly. This is necessary for the effective application of Theorem A.13 on examples.

Case 1. If Ω is a cyclic group Ω then the subgroup L/Q is a cyclic subgroup. Suppose

Ω= {1,g,,gr-1} andL/Q= {1,gd,,gd(p-1)},

where pd=r. Let ζ be a primitive pth root of unity and define an automorphism

σ: HP HP g ζg, Ti Ti, 0in.

The map σ is an algebra isomorphism since it preserves the relations in (4.6) and (4.10). Furthermore, σ gives rise to a /p action on HP and

HL= (HP) /p . (4.15)

Case 2. If the root system R is of type Dn,n even, then Ω/2×/2 and the subgroups of Ω correspond to the intermediate lattices QLP. Suppose

Ω= { 1,g1,g2, g1g2 g12=g22, g1g2=g2g1 } .

and define automorphisms of HP by

σ1: HP HP g1 -g1, g2 g2, Ti Ti, and σ2: HP HP g1 g1, g2 -g2, Ti Ti.


HL1= (HP) σ1 , HL2= (HP) σ2 ,and HQ= (HP)σ1,σ2 , (4.16)

where L1 and L2 are the two intermediate lattices strictly between Q and P.

Notes and References

This is an excerpt of the paper entitled Affine Hecke Algebras, Cyclotomic Hecke algebras and Clifford Theory authored by Arun Ram and Jacqui Ramagge. It was dedicated to Professor C.S. Seshadri on the occasion of his 70th birthday.

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

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