Classification of graded Hecke algebras for complex reflection groups

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 22 January 2014

Examples

4A. The symmetric group G(1,1,n)=Sn

Let V be an n dimensional vector space with orthonormal basis v1,,vn and let Sn act on V by permuting the vi. Let A be a graded Hecke algebra for Sn. Any element which is a product of two reflections is conjugate to (1,2,3) or (1,2)(3,4). The element (1,2)(3,4) has order 2 and so, in the algebra A, [vi,vj]= ki,j ( a(i,j,k) (vi,vj) t(i,j,k)+ a(j,i,k) (vi,vj) t(j,i,k) ) , since vi or vj is in Vg=kerag for all other three cycles g. Since, by (1.6), a(j,i,k) (vi,vj) = a(i,j,k) (vj,vi) = -a(i,j,k) (vi,vj), the graded Hecke algebra A is defined by the relations [vi,vj]=β ki,j ( t(i,j,k)- t(j,i,k) ) andtwvi= vw(i)tw, (4.1) where wSn, 1i,jn, ij, and β=a(1,2,3)(v1,v2).

Let k. Then, with h as in (3.3), vi,h= 12<mk vi,v-vm t(,m)=k2 ( i<t(i,)- i>t(,i) ) =k2isgn (-i)t(i,). (4.2) If fSn, let f|tg denote the coefficient of tg in f. Let A be the graded Hecke algebra defined by the relations in (4.1) with β = a(i,j,) (vi,vj)= [vi,h,vj,h] |t(i,j,) = (k2/4) ( t(i,) t(j,)+ t(i,j) t(i,)- t(j,) t(i,j) ) |t(i,j,) =k2/4. (4.3) If vi=vi-vi,h and si is the simple reflection (i,i+1) then, by Theorem 3.5, vivj = vjvi, tsivi =vi+1 tsi+k,tsi vi+1=vi tsi-k,and tsjvi = vitsj, for|i-j|>1, (4.4) and the algebra A is the graded Hecke algebra Hgr for Sn which is defined in Section 3. When k=1, the map A Sn tw tw vi 12it(i,) (4.5) is a surjective algebra homomorphism.

4B. The hyperoctahedral group G(2,1,n)=WBn

We use the notation from Section 2B so that the group G(2,1,n) is acting by orthogonal matrices on the n dimensional vector space V with orthonormal basis {v1,,vn}. In this case, ξi denotes the diagonal matrix with all ones on the diagonal except for -1 in the (i,i)th entry.

Let A be a graded Hecke algebra for G(2,1,n). If β1=a(i,j,k)(vi,vj) and β2=aξ1(1,2)(v1,v2), then, in the algebra A, [vi,vj]=β2 ( tξ1(1,2)- tξ2(1,2) ) +β1i,j ( t(i,j,)- tξi,ξ(i,j,)- tξi,ξj(i,j,)+ tξjξ(i,j,) +tξiξj(j,i,) +tξjξ(j,i,) -tξiξ(j,i,) -t(j,i,) ) . (4.6)

Let ks,k. Then, with h as in (3.3), vi,h = ks2 vi,2v tξ-k2 <m vi,v-vm t(,m)-k2 <m vi,v+vm tξ(,m) = k2tξi-k2 ( i< ( t(i,)+ tξiξ(i,) ) +i> ( -t(i,)+ tξiξ(i,) ) ) . (4.7) If fG(2,1,n), let f|tg denote the coefficient of tg in f. With notation as in (4.6), let A be the graded Hecke algebra for G(2,1,n) with β1 = a(i,j,) (vi,vj)= [ vi,h, vj,h ] |t(i,j,) = (k2/4) ( t(i,) t(j,)+ t(i,j) t(i,)- t(j,) t(i,j) ) |t(i,j,) =k2/4,and β2 = [ vi,h, vj,h ] |tξi(i,j) = (1/2)ksk ( -tξit(i,j) +t(i,j)tξj -tξj tξiξj(i,j) -tξiξj(i,j) tξi ) |tξi(i,j) =-ksk. If vi=vi-vi,h, then, by Theorem 3.5, the vi commute and the algebra A is the algebra Hgr for WBn defined in Section 3.

4C. The type Dn Weyl group G(2,2,n)=WDn

We shall use the notation from Section 2B so that the group G(2,2,n) is acting by orthogonal matrices on the n dimensional vector space V with orthonormal basis {v1,,vn}. This is an index 2 subgroup of G(2,1,n), and our notation is the same as used above for WBn.

Let A be a graded Hecke algebra for G(2,2,n). If β=a(i,j,k)(vi,vj) then, in the algebra A, [vi,vj]=β i,j ( t(i,j,k)- tξiξ(i,j,)- tξiξj(i,j,)+ tξjξ(i,j,) +tξiξj(j,i,) +tξjξ(j,i,) -tξiξ(j,i,) -t(j,i,) ) . (4.8)

Let k. Then, with h as in (3.3), vi,h= k2 ( i< ( t(i,)+ tξiξ(i,) ) +i> ( -t(i,)+ tξiξ(i,) ) ) . (4.9) If fG(2,2,n), let f|tg denote the coefficient of tg in f. With notation as in (4.8), let A be the graded Hecke algebra for G(2,2,n) with β = a(i,j,) (vi,vj)= [ vi,h, vj,h ] |t(i,j,) = (k2/4) ( t(i,) t(j,)+ t(i,j) t(i,)- t(j,) t(i,j) ) |t(i,j,) =k2/4. If vi=vi-vi,h, then, by Theorem 3.5, the vi commute and the algebra A is the algebra Hgr for WDn defined in Section 3.

4D. The dihedral group I2(r)=G(r,r,2) of order 2r

We shall use the notation for G(r,r,2) from Section 2B so that the group G(r,r,2) is acting by unitary matrices on the 2 dimensional vector space V with orthonormal basis {v1,v2}. The group G(r,r,2) is realized as a real reflection group by using the basis ε1=12 (v1+v2), ε2=-1i2 (v1-v2). This basis is also orthonormal and, with respect to this basis, G(r,r,2) acts by the matrices ( cos(2πm/r) sin(2πm/r) sin(2πm/r) ±cos(2πm/r) ) ,0mr-1.

Let A be a graded Hecke algebra for G(r,r,2). The conjugacy classes of elements which are products of two reflections are {ξ1kξ2-k,ξ1-kξ2k}, 0<k<r/2. Then, in the algebra A, [ε1,ε2]= 0<k<r/2 βk ( tξ1kξ2-k- tξ1-kξ2k ) ,whereβk= aξ1kξ2-k (ε1,ε2). (4.10)

When r is even, there are two conjugacy classes of reflections { ξ12k ξ2-2k (1,2)| 0k<r/2 } and { ξ12k+1 ξ2-(2k+1) (1,2)| 0k<r/2 } . The reflection ξ1mξ2-m(12) is the reflection in the line perpendicular to the vector αm=sin(-2πm/2r) ε1+cos(-2πm/2r) ε2, and the vectors αm can be taken as a root system for G(r,r,2). With h as in (3.3) and ks,k, ε1,h = 0k<r/2 ( kssin(-2k2π/2r) tξ12kξ2-2k(1,2)+ ksin(-(2k+1)2π/2r) tξ12k+1ξ2-(2k+1)(1,2) ) , ε2,h = 0k<r/2 ( kscos(-2k2π/2r) tξ12kξ2-2k(1,2)+ kcos(-(2k+1)2π/2r) tξ12k+1ξ2-(2k+1)(1,2) ) . (4.11) If fG(r,r,2), let f|tg denote the coefficient of tg in f. With notation as in 4.10, let A be the graded Hecke algebra for G(r,r,2) with βk=aξ1kξ2-k (ε1,ε2)= [ε1,h,ε2,h] |tξ1kξ2-k= { sin(k2π/2r) rksk ifkis odd sin(k2π/2r) r2 (ks2+k2) ifkis even. (4.12) If εi=εi-εi,h, then by Theorem 3.5, the εi commute and the algebra A is the algebra Hgr for I2(r) defined in Section 3.

When r is odd, all aspects of the calculation in (4.11) and (4.12) are the same as for the case r even except that there is only one conjugacy class of reflections, {ξ1kξ2-k(1,2)|0kr-1}, and so ks=k.

4E. The group G(r,r/2,2), r/2 odd

We use the notation from Section 2B, or from above for the group G(r,r,2). In this case, the group is not a real reflection group, hence G(r,r/2,2) acts by unitary matrices but not by orthogonal matrices.

Let A be a graded Hecke algebra for G(r,r/2,2). The only conjugacy class for which ag can be nonzero is {tξ1kξ2r/2-k(1,2)|0k<r}. Thus, in the algebra A, [v1,v2]=β k ( tξ12kξ2r/2-2k(1,2)- tξ1r/2-2kξ22k(1,2) ) ,whereβ= aξ2r/2(1,2) (v1,v2).

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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