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

The classification for reflection groups

A reflection is an element of GL(V) that has exactly one eigenvalue not equal to 1. The reflecting hyperplane of a reflection is the (n-1)-dimensional subspace which is fixed pointwise. A complex reflection group G is a finite subgroup of GL(V) generated by reflections. The group G is irreducible if V cannot be written in the form V=V1V2 where V1 and V2 are G-invariant subspaces. The group G is a real reflection group if V=V for a real vector space V and GGL(V).

The following facts about reflection groups are well known.

Let G be an irreducible reflection group.

(a) [STo1954, Theorem 5.3] The number of elements gG such that codim(Vg)=2 is i<jmimj where m1,,mn are the exponents of G.
(b) [Car1972, Lemma 2] If G is a real reflection group and gG with codim(Vg)=2, then g is the product of two reflections.
(c) [OTe1992, Theorem 6.27] For any gG, the space Vg is the intersection of reflecting hyperplanes.

Remark. The statement of Lemma 2.1b does not hold for complex reflection groups. Consider the exceptional complex reflection group G4 of rank 2, in the notation of Shephard and Todd [STo1954]. All the reflections have order 3 and -1G4. Suppose -1=rs for two reflections r and s. If s has eigenvalues 1 and ω, where ω is a primitive cube root of unity, then r-1=-s has eigenvalues -1 and -ω, a contradiction to the assumption that r is a reflection. Thus -1G4 is not a product of two reflections.

Let GGL(V) be a complex reflection group. Let A be a graded Hecke algebra for G and let gG. Let VG={vV|gv=vfor allgG} be the invariants in V.

(a) If g=1 and dimVG1, then ag=0.
(b) If the order of g is 2, then ag=0.

Proof.

(a) Let ,:V×V be a nondegenerate G-invariant Hermitian form on V and write V=VG(VG) where (VG)={vV|v,w=0for allwVG}. Since dim(VG)1 and a1 is skew symmetric, a1 restricted to VG is 0. There is a basis α1,,αk of (VG) and constants ξ1,,ξk, ξi1, such that the reflections s1,,sk given by siv=v+ (ξi-1) v,αi αi,αi αi,forvV, are in G. Equation (1.6) implies that, for any vV, a1(αi,v)= a1(siαi,siv) =a1 ( ξiαi,v+ (ξi-1) v,αi αi,αi αi ) =ξia1(αi,v), since a1(αi,αi)=0 (as a1 is skew symmetric). Since ξi1, a1(αi,v)=0 for 1ik. Thus kera1=V.

(b) Since g2=1, all eigenvalues of g are ±1. If codim(Vg)2, then ag=0 by Theorem 1.9b. If codim(Vg)=2, then g-idVg (-id(Vg)) as a linear transformation on V. By [Ste1964, Theorem 1.5], [Bou1981 V, §5 Ex. 8], the stabilizer, Stab(Vg), of Vg is a reflection subgroup of G and so there is a reflection sStab(Vg) that is the identity on Vg. So sZG(g) and det(s)=det(s)1, where s is s restricted to (Vg). Thus, by Theorem 1.9b, ag=0.

2A. Real reflection groups

If GGL(V) is a real reflection group then V=V and GGL(V), where V is a real vector space. We shall assume that G is irreducible.

Let us recall some basic facts about real reflection groups which can be found in [Hum1990] or [Bou1981]. The action of G on V has fundamental chambers wC indexed by wG. The roots for G are vectors αV such that the reflections in G are the reflections sα in the hyperplanes Hα= {vV|v,α=0}. For each fundamental chamber C, the reflections s1,s2,,sn in the hyperplanes Hα1,Hα2,,Hαn that bound C form a set of simple reflections for G. The simple reflections obtained from a different choice of fundamental chamber wC are ws1w-1,,wsnw-1.

Let GGL(V) be a real reflection group. Let s1,,sn be a set of simple reflections in G and let mij be the order of sisj. Then gG satisfies g21, codim(Vg)=2, and det(h)=1 for all hZG(g) (the conditions in Theorem 1.9c) if and only if g is conjugate to (sisj)k, with0<k< mij2, for some 1i,jn.

Proof.

: Let α and β be two roots such that Vg=HαHβ (see Lemma 2.1c). Then HαHβ has nontrivial intersection with some fundamental chamber C for W, and we may assume that Hα and Hβ are walls of the chamber C (since C is a cone in n). Since choosing simple reflections with respect to a different chamber wC corresponds to conjugation by w, we may assume that the reflections in the hyperplanes Hα and Hβ are simple reflections and α=α1 and β=α2.

The element g is an element of the stabilizer Stab(Vg), which is a reflection group by [Ste1964, Theorem 1.5]. Since codim(Vg)=2, Stab(Vg) is a rank two real reflection group, and therefore a dihedral group. This dihedral group is generated by the two simple reflections s1 and s2 in the hyperplanes Hα1 and Hα2 (restricted to (Vg)) and all reflections have determinant -1. Let g be the element g restricted to (Vg). Since gZG(g), det(g)=1, and so g must be a product of an even number of reflections. Thus g=(s1s2)k or g=(s2s1)k, for some 0<km/2, where m is the order of s1s2. Since g21, km/2, and so g is conjugate to (s1s2)k for some 0<k<m/2.

: Assume that g=(sisj)k for some 0<k<mij/2. Then Vg=HαiHαj and so codim(Vg)=2. Since g is a product of an even number of reflections, det(g)=1. The only elements of O(V)O2() that are diagonalizable in GL(V)GL2() are ±1 and elements with determinant -1. Thus, the eigenvectors of the element g (which has distinct eigenvalues since it is not ±1) do not lie in V, only in V=. Let hZG(g) and let hO(V)O2() denote h restricted to (Vg). Since h commutes with g and g has distinct eigenvalues, g and h have the same eigenvectors. Hence, deth=1.

Using Theorem 2.3 and Theorem 1.9b, we can read off the graded Hecke algebras for the irreducible real reflection groups from the Dynkin diagrams. For each irreducible real reflection group, label a set of simple reflections s1,,sn using the Dynkin diagrams below. If nodes i and j and nodes j and k are connected by single edges, then sisj is conjugate to sjsk via the element sisjsk.

The following table gives representatives of the conjugacy classes of gG that may have ag0 for some graded Hecke algebra A. We assume that the reflection group G is acting on its irreducible reflection representation V. When G is the symmetric group Sn acting on an n-dimensional vector space V by permutation matrices, then dim(VG)=1 and, by Lemma 2.2a and Theorem 2.3, ag0 for some graded Hecke algebra A only if g is conjugate to the three cycle (1,2,3)=s1s2 (this example is analyzed in Section 3). Type Representativeg withag0 An-1 s1s2 Bn s1s2,s2s3 Dn s2s3 E6,E7,E8 s1s4 F4 s1s2,s2s3,s3s4 H3,H4 s1s2,(s1s2)2,s2s3 I2(m) (s1s2)k,0<k<m/2 Table 1.Graded Hecke algebras for real reflection groups. An-1 1 2 n-2 n-1 Bn 1 2 3 n-1 n Dn 1 2 3 4 n-1 n E6 2 3 1 4 5 6 E7 2 3 1 4 5 6 7 E8 2 3 1 4 5 6 7 8 F4 1 2 3 4 H3 1 2 3 5 H4 1 2 3 4 5 I2(m) 1 2 m Figure 1.Coxeter-Dynkin diagrams for real reflection groups.

2B. Complex reflection groups

The irreducible complex reflection groups were classified by Shephard and Todd [STo1954]. There is one infinite family denoted G(r,p,n) and a list of exceptional complex reflection groups denoted G4,,G35. In this subsection, we classify the graded Hecke algebras for the groups G(r,p,n).

Let r, p and n be positive integers with p dividing r and let ξ=e2πi/r. Let Sn be the symmetric group of n×n matrices and let ξj=diag (1,1,,1,ξ,1,,1), where ξ appears in the jth entry. Then G(r,p,n)= { ξ1λ1 ξnλnw| wSn,0λi r-1,λ1+ +λn=0modp } . For λ=(λ1,,λn)(/r)n, let ξλ=ξ1λ1ξnλn. Then the multiplication in G(r,p,n) is described by the relations ξλξμ=ξλ+μ andwξλ=ξwλ w,forλ,μ (/r)n,w Sn, where Sn acts on (/r)n by permuting the factors. Let vi be the column vector with 1 in the ith entry and all other entries 0. The group G(r,p,n) acts on Vn with orthonormal basis {v1,,vn} as a complex reflection group.

Every real reflection group is a complex reflection group and several of these are special cases of the groups G(r,p,n). In particular,

(a) G(1,1,n) is the symmetric group Sn,
(b) G(2,1,n) is the Weyl group WBn of type Bn,
(c) G(2,2,n) is the Weyl group WDn of type Dn, and
(d) G(r,r,2) is the dihedral group I2(r) of order 2r.

The reflections in G(r,p,n) are ξikp, 1in,0k (r/p)-1,and ξikξj-k (i,j), 1i<jn, 0kr-1, where (i,j) is the transposition in Sn that switches i and j.

Conjugacy in G(r,p,n). Each element of G(r,p,n) is conjugate by elements of Sn to a disjoint product of cycles of the form ξiλi ξkλk (i,i+1,,k). By conjugating this cycle by ξi-c ξi+1λi ξi+2λi+λi+1 ξkλi++λk-1 G(r,r,n), we have ξi-c ξkc+λi++λk (i,,k),where c=(k-i)λi+ (k-i-1)λi+1 ++λk-1. If i1,i2,,i denote the minimal indices of the cycles and c1,,c are the numbers c for the various cycles, then after conjugating by ξi1c1 ξi-1c-1 ξi-(c1++c-1) G(r,r,n), each cycle becomes ξkλi++λk (i,,k)except the last, which is ξi-a ξnb(i,,n), where a=c1++c and b=a+λi++λn. If k=n-i+1 is the length of the last cycle, then conjugating the last cycle by ξik-1 ξi+1-1 ξn-1 G(r,r,n) gives ξi-a+k ξnb-k (i,,n). If we conjugate the last cycle by ξipG(r,p,n), we have ξi-a+p ξnb-p (i,,n). In summary, any element g of G(r,p,n) is conjugate to a product of disjoint cycles where each cycle is of the form ξka (i,i+1,,k), 0ar-1, (2.4a) except possibly the last cycle, which is of the form ξia ξnb (i,i+1,,n), with0agcd (p,k)-1, (2.4b) where k=n-i+1 is the length of the last cycle.

Centralizers in G(r,p,n). Let ZG(r,p,n)(g)={hG(r,p,n)|hg=gh} denote the centralizer of gG(r,p,n). Since G(r,p,n) is a subgroup of G(r,1,n), ZG(r,p,n)(g)= ZG(r,1,n)(g) G(r,p,n), for any element gG(r,p,n). Suppose that g is an element of G(r,1,n) which is a product of disjoint cycles of the form ξka(i,,k) and that hG(r,1,n) commutes with g. Conjugation by h effects some combination of the following operations on the cycles of g:

(a) permuting cycles of the same type, ξka(i,,k) and ξmb(j,,m) with b=a and k-i=m-j,
(b) conjugating a single cycle ξka(i,,k) by powers of itself, and
(c) conjugating a single cycle ξka(i,,k) by ξibξkb, for any 0br-1.
Furthermore, the elements of G(r,1,n) which commute with g are determined by how they “rearrange” the cycles of g and a count (see [Mac1995, p. 170]) of the number of such operations shows that if gG(r,1,n) and ma,k is the number of cycles of type ξi+ka(i,i+1,,i+k) for g, then Card(ZG(r,1,n)(g)) =a,k ( ma,k!·kma,k ·rma,k ) . (2.5)

Determining the graded Hecke algebras for G(r,p,n). It follows from Lemma 1.8a that if g=ξia+bξk-a(i,,k), then (Vg) has basis { vk-vk-1, vk-1- vk-2,, vi+1-ξavi } ifb=0,and {vi,,vk} ifb0. Thus, if gG(r,p,n) and codim(Vg)=2, then g is conjugate to one of the following elements: b = ξ1aξ3-a (1,2,3), 0agcd(p,3)-1, c = ξ1a+ ξ2-a(1,2), 0(sor1), d = ξ11 ξ22, 10, 20 (sor1), e = (1,2) ξ3, 0, f = (1,2) ξ3aξ4-a (3,4). It is interesting to note that these elements are also representatives of the conjugacy classes of elements in G(r,p,n) which can be written as a product of two reflections.

We determine conditions on the above elements and on r, p, and n to give nontrivial graded Hecke algebras:

(z) The center of G(r,p,n) is Z(G(r,p,n))= { ξ1ξn |n=0modp } . Since ξ1pξnpZ(G(r,p,n)), it follows that p=r or p=r/2 whenever Z(G(r,p,n)) {±1}= { ξ10ξn0, ξ1r/2 ξnr/2 } .
(b1) If n4, the element ξ1ξ2ξ3ξ4-3ZG(b) and has determinant ξ2 on (Vb)=span-{v3-v2,v2-ξav1}.
(b2) If n=3 and p=0 mod 3, the element ξ1p/3ξ2p/3ξ3p/3ZG(b) and has determinant ξ2p/3 on (Vb).
(c1) If n3, the element ξ1ξ2ξ3-2ZG(c) and has determinant ξ2 on (Vc)=span-{v1,v2}.
(c2) If n=2, p=r/2 and p is odd, the element ξ1p/4ξ2p/4ZG(c) and has determinant ξr/2 on (Vc).
(d1) If n3, the element ξ1ξ3-1ZG(d) and has determinant ξ on (Vd)=span-{v1,v2}.
(d2) If p=r/2, the element ξ1r/2ZG(d) and has determinant ξr/2 on (Vd).
(ef) The elements e and f have order 2.
Thus, it follows from Corollary 1.11, Theorem 1.9b, and Lemma 2.2b that if A is a graded Hecke algebra for G(r,p,n), then ab=0 unless (i)r=1,or (ii)r=2,or (iii)n=3andp0mod3, ac=0 unless (i)r=2andp=1,or (ii)n=2andp=r/2, ad=0 unless p=r,n=2andp0mod2, ae=0 always, and af=0 always. In the remaining cases, one uses the description of ZG(g) given just before (2.5) to check that all elements of ZG(g) have determinant 1 on (Vg). Note that n=3 and p0 mod 3 imply that ab=0 for the elements b=ξ1aξ3-a(1,2,3).

We arrive at the following enumeration of the nontrivial graded Hecke algebras for complex reflection groups. (The tensor product algebra S(V)G always exists and corresponds to the case when all of the skew symmetric forms ag are zero). The table below gives representatives of the conjugacy classes of gG that may have ag0 for some graded Hecke algebra A. Group Representativeg withag0 G(1,1,n)=Sn (1,2,3) G(2,1,n)=WBn,n3 ξ1(1,2),(1,2,3) G(2,2,n)=WDn,n3 (1,2,3) G(r,r,2)=I2(r) ξ1kξ2r-k,0<k<r/2 G(r,r/2,2),r/2odd ξ2r/2(1,2) G(r,r,3),r0mod3 (1,2,3) G(r,r/2,3),r/20mod3,r2 (1,2,3) Table 2.Graded Hecke algebras for the groupsG(r,p,n).

2C. Exceptional complex reflection groups

The irreducible exceptional complex reflection groups G are denoted G4,,G35 in the classification of Shephard and Todd. From Table VII in [STo1954], one sees that the center of G is ±1 only in the cases G4, G12, G24 and G33. By Schur’s lemma, the center of an irreducible complex reflection group consists of multiples of the identity. Thus, by Corollary 1.11, the only exceptional complex reflection groups that could have a nontrivial graded Hecke algebra (i.e., with some ag0) are G4, G12, G24 and G33 (we exclude the real groups). We determine the graded Hecke algebras for these groups using Theorem 1.9b and Lemma 2.2.

The rank 2 group G4 has order 24 and seven conjugacy classes. The following data concerning these conjugacy classes are obtained from the computer software GAP [Sch1995] using the package CHEVIE [GHL1996]. In the following table, ω is a primitive cube root of unity and C(g) denotes the conjugacy class of g. Conjugacy class representatives forG4 Order(g) 1 4 3 6 6 3 2 det(g) 1 1 ω ω ω-1 ω-1 1 |𝒞(g)| 1 6 4 4 4 4 1 |ZG(g)| 24 4 6 6 6 6 24 The elements with determinant 1 and order more than 2 in G4 all have order 4. If g is an element of order 4, then |ZG(g)|=4 and every element of ZG(g) has determinant 1 since ZG(g) is generated by g. Hence, by Theorem 1.9b and Lemma 2.2, ag can be nonzero for a graded Hecke algebra for G4 exactly when g has order 4. Thus, the dimension of the space of parameters for graded Hecke algebras of G4 is 1.

The rank 2 group G12 has order 48. The computer software GAP provides the following information about the conjugacy classes of G12. Conjugacy class representatives forG12 Order(g) 1 2 8 6 8 2 3 4 det(g) 1 -1 -1 1 -1 1 1 1 |𝒞(g)| 1 12 6 8 6 1 8 6 |ZG(g)| 48 4 8 6 8 48 6 8 If g is an element in G12 with order more than 2 and determinant 1, then g has order 3, 4, or 6. Let h be any element of order 8. Then h has determinant -1 and commutes with h2 of order 4. Hence, by Theorem 1.9b, if g has order 4, then ag=0. Let g6 be a representative from the class of elements of order 6. Since |ZG(g6)|=6, ZG(g6) is generated by g6 and hence every element of ZG(g6) has determinant 1. We can choose g62 as a representative for the conjugacy class of elements of order 3. As g6 and g62 commute, g6ZG(g62). But |g6|=6=|ZG(g3)|, so ZG(g62) is generated by g6 and every element of ZG(g3) has determinant 1. Thus, ag can be nonzero for a graded Hecke algebra A for G12 exactly when g has order 3 or 6. Thus, the dimension of the space of parameters of graded Hecke algebras for G12 is 2.

The rank 3 group G24 has order 336. Note that -1G24 since Z(G)={±1}. Up to G-orbits, there are two codimension 2 subspaces, L and M, that are equal to Vg for some gG24 (see [OTe1992, App. C, Table C.5]). Furthermore, Stab(L)A2 and Stab(M)B2. We need only consider elements of order 3 in Stab(L)A2 and of order 4 in Stab(M)B2 (as the rest have order 1 or 2). In G24, there is only one conjugacy class of elements of order 3 and only one conjugacy class of elements of order 4 and determinant 1. The table below (obtained using GAP) records information about these classes. Conjugacy class representatives forG24 Order(g) 3 4 det(g) 1 1 |𝒞(g)| 56 42 |ZG(g)| 6 8 If g has order 3, ZG(g) must contain 1,g,g2, and -1, and hence ZG(g) is generated by these elements since |ZG(g)|=6. Thus all elements of ZG(g) have determinant 1 on (Vg). If g has order 4 and determinant 1, then ZG(g) must contain 1, g, g2, g3, and -1, elements which all have determinant 1 on (Vg). Since |ZG(g)|=8, these elements generate ZG(g) and so every element of ZG(g) has determinant 1 on (VG). Hence, ag can be nonzero for a graded Hecke algebra of G24 exactly when g has order 3 or g has order 4 and determinant 1. Thus, the dimension of the space of parameters for graded Hecke algebras for G24 is 2.

The group G33 is the only exceptional complex reflection group of rank 5. It has order 72·6! and degrees 4, 6, 10, 12, 18. There are 45 reflecting hyperplanes and the corresponding reflections all have order 2. Up to G-orbits, there are two codimension 2 subspaces, L and M, that are equal to Vg for some gG33 (see [OTe1992, App. C, Table C.14]). Furthermore, Stab(L)A1×A1 and Stab(M)A2. We need not consider the case where Vg=L since then g has order 2 and hence ag=0 for any graded Hecke algebra by Proposition 2.2b.

We use a presentation for G33 in six coordinates from [STo1954]: Let V{(x1,x2,x3,x4,x5,x6)|xi} and consider the group generated by order 2 reflections about the hyperplanes H1={x2-x3=0}, H2={x3-x4=0}, H3={x1-x2=0}, H4={x1-ωx2=0}, H5={x1+x2+x3+x4+x5+x6=0}, where ω is a primitive cube root of unity. The fixed point space of this (reducible) group is Y=H1H5={(0,0,0,0,x,-x)|x}, and G33 is just the restriction to Y. Let si be the order 2 reflection about Hi. Let g=s1s3. Then Vg=H1H3 and Stab(Vg)A2. Let h=(s1s3s4)2, the diagonal matrix with diagonal {ω,ω,ω,1,1,1}. Then h acts as ω times the identity on (Vg) as (Vg)-span{x1,x2,x3}. Hence, h commutes with g. But (Vg) has dimension 2 and h has determinant ω21 on (Vg). Thus, by Theorem 1.9b and Lemma 2.2, ag=0 for any graded Hecke algebra. The same argument applied to Y shows that G33 has no nontrivial graded Hecke algebras. In summary, the dimension of the space of parameters for graded Hecke algebras for G33 is zero. Group gwithag0 G4 Order(g)=4 G12 Order(g1)=3andOrder(g2)=6 G24 Order(g1)=3andOrder(g2)=4,det(g2)=1 Table 3.Graded Hecke algebras for exceptional complex reflection groups.

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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