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

Graded Hecke algebras

In this section, we define the graded Hecke algebra following Drinfeld [Dri1986]. Our main result in this section is Theorem 1.9c, which determines exactly how many degrees of freedom one has in defining a graded Hecke algebra.

Let V be an n dimensional vector space over and let G be a finite subgroup of GL(V). The group algebra of G is G=-span {tg|gG} ,withtgth= tgh. Let ag:V×V be skew symmetric bilinear forms indexed by the elements of G and let A be the associative algebra generated by V and G with the additional relations thvth-1=hv and[v,w]= gGag (v,w)tg, forhGandv,w V, (1.1) where [v,w]=vw-wv. These relations allow every element aA to be written in the form a=gGpg tg,pgS(V), (1.2) where S(V) is the symmetric algebra of V. More precisely, one must fix a section of the canonical surjection T(V)S(V) from the tensor algebra of V to S(V) and take the elements pg to be in the image of this section.

The structure of A depends on the choices of the “parameters” ag(v,w). Our goal is to determine when the algebra A will be a “semidirect product” of S(V) and G. This idea motivates the following definition [Dri1986].

The algebra A is a graded Hecke algebra for G if AS(V)G as a vector space, or, equivalently, if the expression in (1.2) is unique for each aA. A general element aA is a linear combination of products of elements tg and ui, where {u1,u2,,un} is a basis of V. There are two straightening operations needed to put a in the form (1.2): (a) movingth's to the right,and (b) puttinguiuj pairs in the correct order. These two straightening operations correspond to the two identities in (1.1). Note that the “correct order” of uiuj is determined by the choice of the section of the projection T(V)S(V). Let v1,v2,v3 be arbitrary elements of V and let hG. Applying the straightening operations to thv1v2 gives thv1v2 = th[v1,v2] +thv2v1 (rearrangev1and v2) = th[v1,v2] +(hv2) (hv1)th (movethto the right), and applying the straightening operations in a different order gives thv1v2 = (hv1)(hv2) th (movethto the right) = [hv1,hv2] th+(hv2) (hv1)th. Setting these two expressions equal gives the relation th[v1,v2] th-1= [hv1,hv2], for allhG, v1,v2V. (1.3) Similarly, applying the straightening operations to v1v2v3 gives v1v2v3 = [v1,v2]v3 +v2v1v3 (movingv1pastv2) = [v1,v2]v3+ v2[v1,v3]+ v2v3v1 (movingv1pastv3) = [v1,v2]v3+ v2[v1,v3]+ [v2,v3]v1+ v3v2v1 (straighteningv2pastv3), and applying the straightening operations in a different order gives v1v2v3 = v1[v2,v3]+ v1v3v2 (movingv2pastv3) = v1[v2,v3]+ [v1,v3]v2+ v3v1v2 (movingv1pastv3) = v1[v2,v3]+ [v1,v3]v2+ v3[v1,v2]+ v3v2v1 (straighteningv1pastv2). These are equal if [v1,[v2,v3]]+ [v2,[v3,v1]]+ [v3,[v1,v2]]= 0,for allv1,v2, v3V. (1.4) Conversely, the identities (1.3) and (1.4) are exactly what is needed to guarantee that any order of application of the straightening operations (a) and (b) will produce the same normal form (1.2) for the element a. Thus we have

Let A be an algebra defined as in (1.1). Then A is a graded Hecke algebra if and only if the identities (1.3) and (1.4) hold in A.

Using (1.1), the relations (1.3) and (1.4) can be rewritten in terms of the bilinear forms ag:V×V as ag(v1,v2)= ahgh-1 (hv1,hv2) and (1.6) ag(v3,v1) (gv2-v2)+ ag(v2,v3) (gv1-v1)+ ag(v1,v2) (gv3-v3)=0 (1.7) for v1,v2,v3V and g,hG.

Let ,:V×V be a G-invariant nondegenerate Hermitian form on V. For each gG, set Vg = {vV|gv=v}, (Vg) = { vV|v,w =0for allwVg } ,and kerag = { vV|ag (v,w)=0for all wV } .

Let G be a finite subgroup of GL(V) and let gG.

(a) (Vg)={v-gv|vV}.
(b) Suppose g1. If codim(Vg)=2 and a:V×V is a skew symmetric bilinear form such that kera=Vg, then a satisfies (1.7).
Let A be a graded Hecke algebra for G defined by skew symmetric bilinear forms ag:V×V.
(c) If g1 then keragVg.
(d) If g1 and ag0 then kerag=Vg and codim(Vg)=2.
(e) If g1 and ag0 then, for all hG, ah-1gh (b1,b2)= det(h)ag (b1,b2), where {b1,b2} is a basis of (Vg) and h:(Vg)(Vg) is the composition of h restricted to (Vg) with the canonical projection VV/Vg.


(a) Consider the map ϕ:VV given by ϕ(v)=v-gv. Then kerϕ=Vg and imϕ(Vg) since, if vV, wVg, then v-gv,w= v,w- gv,w= v,w- gv,gw= v,w- v,w. Since dim(imϕ)=codim(kerϕ)=codim(Vg) it follows that imϕ=(Vg).

(b) Let v1,v2,v3V. If any viVg, then (1.7) holds trivially for the skew symmetric form a. So assume each viVg and write each vi as vi++vi- where vi+Vg and vi-(Vg). Then a(vi,vj)= a(vi-,vj-) andvi-gvi= vi--gvi-. Since dim(Vg)=2, at least one of the vi- is a linear combination of the other two. Say vi-=c2v2-+c3v3- with c2,c3. Substituting vi-gvi=vi--gvi- and v1-=c2v2-+c3v3- then yields a(v3,v1) (gv2-v2)+ a(v2,v3) (gv1-v1)+ a(v1,v2) (gv3-v3) = a(v3-,v1-) (gv2--v2-)+ a(v2-,v3-) (gv1--v1-)+ a(v1-,v2-) (gv3--v3-)= 0, and so (1.7) holds.

(c) Let v3Vg and v2V.
If v2Vg, then ag(v2,v3)(gv1-v1)=0 for all v1V by (1.7). Since VgV, there is some v1 such that gv1v1 and so ag(v2,v3)=0.
If v2Vg, let v1=k=1r-1gkv2, where r is the order of g. By (1.6), ag(v3,gkv2)= ag(g-kv3,v2)= ag(v3,v2), for any k, and so 0 = ag(v3,v1) (gv2-v2)+ ag(v2,v3) (gv1-v1) = (r-1) ag(v3,v2) (gv2-v2)+ ag(v3,v2) (gv2-v2)=r ag(v3,v2) (gv2-v2). Thus ag(v3,v2)=0.
Hence, for all v2V, ag(v3,v2)=0 and so Vgkerag.

(d) By (c), codim(Vg)codim(kerag). Since ag0, there exist v,wV with ag(v,w)0 and so codim(kerag)2. Let v1-gv1 and v2-gv2 be linearly independent elements of (Vg). Then (1.7) implies that any element v3-gv3(Vg) is a linear combination of v1-gv1 and v2-gv2, and so 2dim((Vg)) =codim(Vg)codim (kerag)2. Thus Vg=kerag and codim(Vg)=2.

(e) Write hb1=h11b1+h21b2+(hb1)g and hb2=h12b1+h22b2+(hb2)g with hij and (hbi)gVg. Then ah-1gh (b1,b2) = ag(hb1,hb2) =ag(h11b1+h21b2+(hb1)g,h12b1+h22b2+(hb2)g) = (h11h22-h21h12) ag(b1,b2)=det (h)ag(b1,b2) since ag is skew symmetric and Vgkerag.

The following theorem is a slightly strengthened version of statements (given without proof) in [Dri1986].

Let G be a finite subgroup of GL(V) and let ZG(g)={hG|hg=gh} denote the centralizer of an element g in G.

(a) If A is a graded Hecke algebra for G, then the values of ah-1gh are determined by the values of ag via the equation ah-1gh (v1,v2)= ag(hv1,hv2) ,for allg,hG, v1,v2V.
(b) For g1, there is a graded Hecke algebra A with ag0 if and only if kerag=Vg, codim(Vg)=2, anddet(h) =1,for all hZG(g), where h is h restricted to the space (Vg). In this case, ag is determined by its value ag(b1,b2) on a basis {b1,b2} of (Vg).
(c) Let d be the number of conjugacy classes of gG such that codim(Vg)=2 and det(h)=1 for all hZG(g), where h is h restricted to the space (Vg). The sets {ag}gG corresponding to graded Hecke algebras A form a vector space of dimension d+dim((2V)G).


(a) is simply a restatement of (1.6).

(b) : If A is a graded Hecke algebra and ag0 then by Lemma 1.8d, codim(Vg)=2 and kerag=Vg. So ag is determined by its value ag(b1,b2) on a basis b1,b2 of (Vg). Suppose hZG(g). Then, by Lemma 1.8e, ag(b1,b2)= ahgh-1 (hb1,hb2)= ag(hb1,hb2) =det(h)ag (b1,b2), and so det(h)=1. Note that h(Vg)=Vg and h(Vg)=(Vg) since, for each vVg, h(v)=hg(v)=gh(v).

: If codim(Vg)=2 then, up to constant multiples, there is a unique skew symmetric form on V which is nondegenerate on (Vg) and which has kerag=Vg. Fix such a form and then define forms ah, hG, by ah(v1,v2)= { ag(kv1,kv2) ifh=k-1gk, 0 otherwise, (1.10) for v1,v2V. Let a1 be any G-invariant skew symmetric form on V. Then this collection {ag}gG of skew symmetric bilinear forms satisfies (1.6) by definition and (1.7) by Lemma 1.8b. Thus (by Lemma 1.5), it determines a graded Hecke algebra A via (1.1).

(c) From (a) and (b) it follows that the sets {ag}gG, running over all graded Hecke algebras A for G, form a vector space. Since each of the collections {ag}g1 constructed by (1.10) has its support on a single conjugacy class, these collections form a basis of the vector space of sets {ag}g1. The only condition on the form a1 is that it satisfies (1.6), which means that it is a G-invariant element of (2V)*.

The following consequence of Theorem 1.9 will be useful for completing the classification of graded Hecke algebras for complex reflection groups.

Assume that G contains h=ξ·1 for some ξ\{±1}. If A is a graded Hecke algebra for G, then ag=0 for all g1.


If h=ξ·1G, then hZG(g) for every gG and det(h)=ξ2 if codim(Vg)=2. The statement then follows from Theorem 1.9b.

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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