Classification of graded Hecke algebras for complex reflection groups

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

Last update: 22 January 2014

The graded Hecke algebras Hgr

In [Lus1989], Lusztig gives a definition of graded Hecke algebras for real reflection groups which is different from the definition in Section 1, which applies to more general groups. It is not obvious that Lusztig’s algebras are examples of the graded Hecke algebras defined in Section 1. In this section, we show explicitly how the definition of Section 1 includes Lusztig’s algebras.

Let W be a finite real reflection group acting on V and let R be the root system of W. Let α1,,αn be a choice of simple roots in R and let s1,,sn be the corresponding simple reflections in W. Let sα be the reflection in the root α so that, for vV, sαv=v-v,α α,whereα=2α /α,α. Let R+={α>0} denote the set of positive roots in R.

Let kα be fixed complex numbers indexed by the roots αR satisfying kwα=kα, for allwW,αR. (3.1) This amounts to a choice of either one or two “parameters”, depending on whether all roots in R are the same length or not. As in Section 1, let W=-span{tg|gW}, with tgth=tgh, and let S(V) be the symmetric algebra of V. Lusztig [Lus1989] defines the “graded Hecke algebra” with parameters {kα} to be the unique algebra structure Hgr on the vector space S(V)W such that

(3.2 a) S(V)=S(V)1 is a subalgebra of Hgr,
(3.2 b) W=1W is a subalgebra of Hgr, and
(3.2 c) tsiv=(siv)tsi-kαiv,αi, for all vV and simple reflections si in the simple roots αi.
We shall show that every algebra Hgr as defined by (3.2 a-c) is a graded Hecke algebra A for a specific set of skew symmetric bilinear forms ag.

Let kα as in (3.1). Use the notation h=12α>0 kααtsα ,so thatv,h =12α>0 kαv,α tsα (3.3) for vV. The element h should be viewed as an element of VW, and v,hW. With this notation, let A be the algebra (as in Section 1) generated by V and W with relations tgv=(gv)tg and[v,w]=- [v,h,w,h], forv,wV,gW. (3.4) Note that A is defined by the bilinear forms ag(v,w)=14 α,β>0g=sαsβ kαkβ ( v,β w,α- v,α w,β ) .

The following theorem shows that the algebra A satisfies the defining conditions (3.2 a-c) of the algebra Hgr.

Let W be a finite real reflection group and let A be the algebra defined by (3.4).

(a) As vector spaces, AS(V)W (and hence, A is a graded Hecke algebra).
(b) If v=v-v,h for vV, then [v,w]=0 andtsi v=(siv˜) tsi-kαi v,αi, for all v,wV and simple reflections si in W.


First note that if u,vV then [u,v,h]= 12α>0kα v,α u,αα tsα= [v,u,h]. (*) Thus, for u,v,wV, [u,[v,w]]+ [w,[u,v]]+ [v,[w,u]] = [u,[w,h,v,h]]+ [w,[v,h,u,h]]+ [v,[u,h,w,h]] = [[u,w,h],v,h]+ [w,h,[u,v,h]]+ [[w,v,h],u,h] +[v,h,[w,u,h]] +[[v,u,h],w,h] +[u,h,[v,w,h]] = [[w,u,h],v,h]+ [w,h,[v,u,h]]+ [[v,w,h],u,h] +[v,h,[w,u,h]] +[[v,u,h],w,h] +[u,h,[v,w,h]] = 0. (3.6) For vV, hW, and si a simple reflection, tsiv,h tsi = 12α>0 kαv,α tssiα= ( 12α>0kα v,siα tsα ) +kαi v,αi tsi = ( 12α>0kα siv,α tsα ) +kαi v,αi tsi=siv,h +kαiv,αi tsi. (3.7) Using this equality, we obtain tsi[v,w]tsi = -tsi [v,h,w,h] tsi = - [ siv,h+ kαiv,αi tsi,siw,h +kαiw,αi tsi ] = [siv,siw]- kαiv,αi [tsi,siw,h] -kαiw,αi [siv,h,tsi] = [siv,siw]- kαiv,αi ( tsisiw,h tsi-siw,h ) tsi +kαiw,αi ( tsisiv,h tsi-siv,h ) tsi = [siv,siw]- kαiv,αi ( w,htsi+ kαisiw,αi -siw,htsi ) +kαiw,αi ( v,htsi+ kαisiv,αi -siv,htsi ) = [siv,siw]- kαiv,αi w,αi αi,htsi -kαi2v,αi w,siαi +kαiw,αi v,αi αi,htsi +kαi2w,αi v,siαi = [siv,siw]. (3.8) The two identities (3.6) and (3.8), as in (1.3) and (1.4), show that the algebra A is isomorphic to S(V)W.

(b) This can now be proved by direct computation. If v,wV then [v,w]= [ v-v,h, w-w,h ] =[v,w]+ [v,h,w,h] -[v,w,h] +[w,v,h]=0, by equation (3.4) and equation (*) in the proof of Theorem 3.5. If vV and si is a simple reflection then, by (3.7), tsivtsi= tsivtsi-tsi v,htsi= siv-siv,h -kαiv,αi tsi=siv˜- kαiv,αi tsi.

Theorem 3.5b shows that if A is the graded Hecke algebra defined by (3.4), then the elements v, for vV, generate a subalgebra of A isomorphic to S(V) and these elements together with the tsi satisfy the relations of (3.2 c). Since part (a) of Theorem 3.5 shows that A is isomorphic to S(V)W as a vector space, it follows that A satisfies the conditions (3.2 a-c), relations which uniquely define the graded Hecke algebra A. Thus, Lusztig’s algebras are special cases of the graded Hecke algebras defined in Section 1. Furthermore, by comparing the dimensions of the parameter spaces, we see that there are graded Hecke algebras that are not isomorphic to algebras defined by Lusztig for the Coxeter groups F4, H3, H4, and I2(m).

Notes and references

This is a typed version of Classification of graded Hecke algebras for complex reflection groups by Arun Ram and Anne V. Shepler.

Research of the first author supported in part by the National Security Agency and by EPSRC Grant GR K99015 at the Newton Institute for Mathematical Sciences. Research of the second author supported in part by National Science Foundation grant DMS-9971099.

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