Classification of graded Hecke algebras for complex reflection groups

Arun Ram Department of Mathematics University of Wisconsin, Madison Madison, WI 53706 USA

                                ram@math.wisc.edu

and

Anne V. Shepler Department of Mathematics University of North Texas Denton, TX 76203 USA

                                 ashepler@unt.edu

Abstract.

The graded Hecke algebra for a finite Weyl group is intimately related to the geometry of the Springer correspondence. A construction of Drinfeld produces an analogue of a graded Hecke algebra for any finite subgroup of GL(V). This paper classifies all the algebras obtained by applying Drinfeld’s construction to complex reflection groups. By giving explicit (though nontrivial) isomorphisms, we show that the graded Hecke algebras for finite real reflection groups constructed by Lusztig are all isomorphic to algebras obtained by Drinfeld’s construction. The classification shows that there exist algebras obtained from Drinfeld’s construction which are not graded Hecke algebras as defined by Lusztig for real as well as complex reflection groups.

0. Introduction

This paper is motivated by a general effort to generalize the theory of Weyl groups and their relation to groups of Lie type to the setting of complex reflection groups. One natural question is whether there are affine Hecke algebras corresponding to complex reflection groups. If they exist then it might be possible to use these algebras to build an analogue of the Springer correspondence for complex reflection groups.

A priori, one knows how to construct affine Hecke algebras corresponding only to Weyl groups since both a finite real reflection group W and a W -invariant lattice (the existence of which forces W to be a Weyl group) are needed in the construction. Our search for analogues of graded Hecke algebras for complex reflection groups was motivated by Lusztig’s results [Lu2] showing that the geometric information contained in the affine Hecke algebra can be recovered from the corresponding graded Hecke algebra. Lusztig [Lu] defines the graded Hecke algebra for a finite Weyl group W with reflection representation V. Let t_{g}, g ∈ W, be a basis for the group algebra ℂW of W and let k_{α} ∈ ℂ be “parameters” indexed by the roots in the root system of W such that k_{α} depends only on the length of the root α. Then the graded Hecke algebra H_{gr} depending on the parameters k_{α} is the (unique) algebra structure on S(V) ⊗ ℂW such that

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.

Keywords: reflection group, Coxeter group, Weyl group, affine Hecke algebra, Iwahori-Hecke algebra, representation theory, graded Hecke algebra.

(a) the symmetric algebra of V, S(V) = S(V) ⊗ 1, is a subalgebra of H_{gr},

(b) the group algebra ℂW = 1 ⊗ ℂW = span-{1 ⊗ t_{g} | g ∈ W } is a subalgebra of H_{gr}, and (c) t_{s_{i}}v = (s_{i}v)t_{s_{i}} − k_{α_{i}}⟨v, α_{i}^{∨}⟩ for all v ∈ V and simple reflections s_{i} in the simple roots α_{i}.

This definition applies to all finite real reflection groups W since the simple roots and simple reflections are well defined. Unfortunately, the need for simple reflections in the construction makes it unclear how to define analogues for complex reflection groups.

For finite real reflection groups, the graded Hecke algebra H_{gr} is a “semidirect product” of the polynomial ring S(V) and the group algebra ℂW. Drinfeld [Dr] defines a different type of semidirect product of S(V) and ℂW, and Drinfeld’s construction applies to all finite subgroups G of GL(V). In this paper, we

(1) Classify all the algebras obtained by applying Drinfeld’s construction to finite complex reflection groups G,

(2) Show that every graded Hecke algebra H_{gr} (as defined by Lusztig) for a finite real reflection group is isomorphic to an algebra obtained by Drinfeld’s construction by giving explicit isomorphisms between these algebras.

The results from (2) show how Drinfeld’s construction is a true generalization of Lusztig’s construction of graded Hecke algebras, something which is not obvious. Our classification in (1) gives a complete solution to the problem of finding all graded Hecke algebras for finite reflection groups.

A consequence of our classification is that there exist graded Hecke algebras for finite real reflection groups which are not obtained with Lusztig’s construction. In this sense, Drinfeld’s construction is a strict generalization of the algebras H_{gr}. These new algebras, and the algebras corresponding to complex reflection groups that are not real reflection groups, deserve further study and probably have interesting representation theories.

For us, one surprising result of our classification is that no nontrivial graded Hecke algebra structures exist for many complex reflection groups. In some sense, this is disappointing, as we would have liked to find nontrivial and interesting structures for each complex reflection group.

It might be that we have not yet hit upon the “right” definition of graded Hecke algebras. For example, we show that there do not exist nontrivial graded Hecke algebra structures, according to Drinfeld’s definition, for any of the complex reflection groups G(r, 1, n) = (ℤ/rℤ) ≀ S_{n} when r > 2 and n > 3. On the other hand, in the last section of this paper we are able to construct algebras that “look” like they ought to be graded Hecke algebras corresponding to these groups. Is it possible that there is a “better” definition of graded Hecke algebras which applies to complex reflection groups and which includes the algebras that we introduce in Section 5 as examples?

Acknowledgements. We thank C. Kriloff for numerous stimulating conversations during our work on this paper. A. Ram thanks the Newton Institute for the Mathematical Sciences at Cambridge University for hospitality and support (EPSRC Grant No. GR K99015) during Spring 2001 when the writing of this paper was completed.

1. Graded Hecke algebras

In this section, we define the graded Hecke algebra following Drinfeld [Dr]. 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 ℂ and let G be a finite subgroup of GL(V). The group algebra of G is

ℂG = ℂ-span{t_{g} | g ∈ G}, with t_{g}t_{h} = t_{gh}.

Let a_{g}: V × V ⟶ ℂ be skew symmetric bilinear forms indexed by the elements of G and let A be the associative algebra generated by V and ℂG with the additional relations

t_{h}vt_{h^{−1}} = hv  and  [v, w] = ∑_{g∈G} a_{g}(v, w) t_{g},  for h ∈ G and v, w ∈ V,  (1.1)

where [v, w] = vw − wv. These relations allow every element a ∈ A to be written in the form

a = ∑_{g∈G} p_{g}t_{g},  p_{g} ∈ S(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 p_{g} to be in the image of this section.

The structure of A depends on the choices of the “parameters” a_{g}(v, w) ∈ ℂ. Our goal is to determine when the algebra A will be a “semidirect product” of S(V) and ℂG. This idea motivates the following definition [Dr].

The algebra A is a graded Hecke algebra for G if

A ≅ S(V) ⊗ ℂG as a vector space,

or, equivalently, if the expression in (1.2) is unique for each a ∈ A. A general element a ∈ A is a linear combination of products of elements t_{g} and u_{i}, where {u_{1}, u_{2}, …, u_{n}} is a basis of V. There are two straightening operations needed to put a in the form (1.2):

(a) moving t_{h}’s to the right,  and  (b) putting u_{i}u_{j} pairs in the correct order.

These two straightening operations correspond to the two identities in (1.1). Note that the “correct order” of u_{i}u_{j} is determined by the choice of the section of the projection T (V) → S(V). Let v_{1}, v_{2}, v_{3} be arbitrary elements of V and let h ∈ G. Applying the straightening operations to t_{h}v_{1}v_{2} gives

t_{h}v_{1}v_{2} = t_{h}[v_{1}, v_{2}] + t_{h}v_{2}v_{1} = t_{h}[v_{1}, v_{2}] + (hv_{2})(hv_{1})t_{h} (rearrange v_{1} and v_{2})

(move t_{h} to the right),

and applying the straightening operations in a different order gives

\begin{aligned}t_{h}v_{1}v_{2} & = (hv_{1})(hv_{2})t_{h} \\ & = [hv_{1}, hv_{2}]t_{h} + (hv_{2})(hv_{1})t_{h}. \end{aligned}  (move t_{h} to the right)

Setting these two expressions equal gives the relation

t_{h}[v_{1}, v_{2}]t_{h^{−1}} = [hv_{1}, hv_{2}],  for all h ∈ G, v_{1}, v_{2} ∈ V .  (1.3)

Similarly, applying the straightening operations to v_{1}v_{2}v_{3} gives

v_{1}v_{2}v_{3} = [v_{1}, v_{2}]v_{3} + v_{2}v_{1}v_{3} = [v_{1}, v_{2}]v_{3} + v_{2}[v_{1}, v_{3}] + v_{2}v_{3}v_{1} = [v_{1}, v_{2}]v_{3} + v_{2}[v_{1}, v_{3}] + [v_{2}, v_{3}]v_{1} + v_{3}v_{2}v_{1} (moving v_{1} past v_{2})

(moving v_{1} past v_{3})
(straightening v_{2} and v_{3}),

and applying the straightening operations in a different order gives

v_{1}v_{2}v_{3} = v_{1}[v_{2}, v_{3}] + v_{1}v_{3}v_{2} = v_{1}[v_{2}, v_{3}] + [v_{1}, v_{3}]v_{2} + v_{3}v_{1}v_{2} = v_{1}[v_{2}, v_{3}] + [v_{1}, v_{3}]v_{2} + v_{3}[v_{1}, v_{2}] + v_{3}v_{2}v_{1} (moving v_{2} past v_{3})

(moving v_{1} past v_{3})
(straightening v_{1} and v_{2}).

These are equal if

[v_{1}, [v_{2}, v_{3}]] + [v_{2}, [v_{3}, v_{1}]] + [v_{3}, [v_{1}, v_{2}]] = 0,  for all v_{1}, v_{2}, v_{3} ∈ V.  (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

Lemma 1.5. 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 a_{g}: V × V → ℂ as

a_{g}(v_{1}, v_{2}) = a_{hgh^{−1}}(hv_{1}, hv_{2})  and  (1.6)
a_{g}(v_{3}, v_{1})(gv_{2} − v_{2}) + a_{g}(v_{2}, v_{3})(gv_{1} − v_{1}) + a_{g}(v_{1}, v_{2})(gv_{3} − v_{3}) = 0  (1.7)

for v_{1}, v_{2}, v_{3} ∈ V and g, h ∈ G.

Let ⟨ , ⟩ : V × V → ℂ be a G-invariant nondegenerate Hermitian form on V. For each g ∈ G, set

V^{g} = {v ∈ V | gv = v},
(V^{g})^{⊥} = {v ∈ V | ⟨v, w⟩ = 0 for all w ∈ V^{g}},  and
ker a_{g} = {v ∈ V | a_{g}(v, w) = 0 for all w ∈ V }.

Lemma 1.8. Let G be a finite subgroup of GL(V) and let g ∈ G.

(a) (V^{g})^{⊥} = {v − gv | v ∈ V }.

(b) Suppose g ≠ 1. If codim(V^{g}) = 2 and a: V × V → ℂ is a skew symmetric bilinear form such that ker a = V^{g}, then a satisfies (1.7).

Let A be a graded Hecke algebra for G defined by skew symmetric bilinear forms a_{g}: V × V → ℂ.

(c) If g ≠ 1 then ker a_{g} ⊇ V^{g}.

(d) If g ≠ 1 and a_{g} ≠ 0 then ker a_{g} = V^{g} and codim(V^{g}) = 2.

(e) If g ≠ 1 and a_{g} ≠ 0 then, for all h ∈ G,

a_{h^{−1}gh}(b_{1}, b_{2}) = det(h^{⊥})a_{g}(b_{1}, b_{2}),

where {b_{1}, b_{2}} is a basis of (V^{g})^{⊥} and h^{⊥}: (V^{g})^{⊥} → (V^{g})^{⊥} is the composition of h restricted to (V^{g})^{⊥} with the canonical projection V → V/V^{g}.

Proof. (a) Consider the map ϕ: V → V given by ϕ(v) = v −gv. Then ker ϕ = V^{g} and im ϕ ⊆ (V^{g})^{⊥} since, if v ∈ V, w ∈ V^{g}, then

⟨v − gv, w⟩ = ⟨v, w⟩ − ⟨gv, w⟩ = ⟨v, w⟩ − ⟨gv, gw⟩ = ⟨v, w⟩ − ⟨v, w⟩.

Since dim(im ϕ) = codim(ker ϕ) = codim(V^{g}) it follows that im ϕ = (V^{g})^{⊥}.

(b) Let v_{1}, v_{2}, v_{3} ∈ V. If any v_{i} ∈ V^{g}, then (1.7) holds trivially for the skew symmetric form a. So assume each v_{i} ∉ V^{g} and write each v_{i} as v_{i}^{+} + v_{i}^{−} where v_{i}^{+} ∈ V^{g} and v_{i}^{−} ∈ (V^{g})^{⊥}. Then

a(v_{i}, v_{j}) = a(v_{i}^{−}, v_{j}^{−})  and  v_{i} − gv_{i} = v_{i}^{−} − gv_{i}^{−}.

Since dim(V^{g})^{⊥} = 2, at least one of the v_{i}^{−} is a linear combination of the other two. Say v_{1}^{−} = c_{2}v_{2}^{−} + c_{3}v_{3}^{−} with c_{2}, c_{3} ∈ ℂ. Substituting v_{i} − gv_{i} = v_{i}^{−} − gv_{i}^{−} and v_{1}^{−} = c_{2}v_{2}^{−} + c_{3}v_{3}^{−} then yields

\begin{matrix}a(v_{3}, v_{1}) & (gv_{2} − v_{2}) + a(v_{2}, v_{3})(gv_{1} − v_{1}) + a(v_{1}, v_{2})(gv_{3} − v_{3}) \\ = a(v_{3}^{−}, v_{1}^{−})(gv_{2}^{−} − v_{2}^{−}) + a(v_{2}^{−}, v_{3}^{−})(gv_{1}^{−} − v_{1}^{−}) + a(v_{1}^{−}, v_{2}^{−})(gv_{3}^{−} − v_{3}^{−}) = 0, \end{matrix}

and so (1.7) holds.

(c) Let v_{3} ∈ V^{g} and v_{2} ∈ V. If v_{2} ∈ V^{g}, then a_{g}(v_{2}, v_{3})(gv_{1} − v_{1}) = 0 for all v_{1} ∈ V by (1.7). Since V^{g} ≠ V, there is some v_{1} such that gv_{1} ≠ v_{1} and so a_{g}(v_{2}, v_{3}) = 0.

If v_{2} ∉ V^{g}, let v_{1} = ∑_{k=1}^{r−1} g^{k}v_{2}, where r is the order of g. By (1.6), a_{g}(v_{3}, g^{k}v_{2}) = a_{g}(g^{−k}v_{3}, v_{2}) =

a_{g}(v_{3}, v_{2}), for any k, and so

\begin{aligned}0 & = a_{g}(v_{3}, v_{1})(gv_{2} − v_{2}) + a_{g}(v_{2}, v_{3})(gv_{1} − v_{1}) \\ & = (r − 1)a_{g}(v_{3}, v_{2})(gv_{2} − v_{2}) + a_{g}(v_{3}, v_{2})(gv_{2} − v_{2}) = ra_{g}(v_{3}, v_{2})(gv_{2} − v_{2}). \end{aligned}

Thus a_{g}(v_{3}, v_{2}) = 0. Hence, for all v_{2} ∈ V, a_{g}(v_{3}, v_{2}) = 0 and so V^{g} ⊆ ker a_{g}.

(d) By (c), codim(V^{g}) ≥ codim(ker a_{g}). Since a_{g} ≠ 0, there exist v, w ∈ V with a_{g}(v, w) ≠ 0 and so codim(ker a_{g}) ≥ 2. Let v_{1} − gv_{1} and v_{2} − gv_{2} be linearly independent elements of (V^{g})^{⊥}. Then (1.7) implies that any element v_{3} − gv_{3} ∈ (V^{g})^{⊥} is a linear combination of v_{1} − gv_{1} and v_{2} − gv_{2}, and so

2 ≥ dim((V^{g})^{⊥}) = codim(V^{g}) ≥ codim(ker a_{g}) ≥ 2.

Thus V^{g} = ker a_{g} and codim(V^{g}) = 2.

(e) Write hb_{1} = h_{11}b_{1} + h_{21}b_{2} + (hb_{1})^{g} and hb_{2} = h_{12}b_{1} + h_{22}b_{2} + (hb_{2})^{g} with h_{ij} ∈ ℂ and (hb_{i})^{g} ∈ V^{g}. Then

\begin{aligned}a_{h^{−1}gh}(b_{1}, b_{2}) & = a_{g}(hb_{1}, hb_{2}) = a_{g}(h_{11}b_{1} + h_{21}b_{2} + (hb_{1})^{g}, h_{12}b_{1} + h_{22}b_{2} + (hb_{2})^{g}) \\ & = (h_{11}h_{22} − h_{21}h_{12})a_{g}(b_{1}, b_{2}) = det(h^{⊥})a_{g}(b_{1}, b_{2}) \end{aligned}

since a_{g} is skew symmetric and V^{g} ⊆ ker a_{g}.

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

Theorem 1.9. Let G be a finite subgroup of GL(V) and let Z_{G}(g) = {h ∈ G | 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 a_{h^{−1}gh} are determined by the values of a_{g} via the equation

a_{h^{−1}gh}(v_{1}, v_{2}) = a_{g}(hv_{1}, hv_{2}), for all g, h ∈ G, v_{1}, v_{2} ∈ V .

(b) For g ≠ 1, there is a graded Hecke algebra A with a_{g} ≠ 0 if and only if

ker a_{g} = V^{g}, codim(V^{g}) = 2, and det(h^{⊥}) = 1, for all h ∈ Z_{G}(g),

where h^{⊥} is h restricted to the space (V^{g})^{⊥}. In this case, a_{g} is determined by its value a_{g}(b_{1}, b_{2}) on a basis {b_{1}, b_{2}} of (V^{g})^{⊥}.

(c) Let d be the number of conjugacy classes of g ∈ G such that codim(V^{g}) = 2 and det(h^{⊥}) = 1 for all h ∈ Z_{G}(g), where h^{⊥} is h restricted to the space (V^{g})^{⊥}. The sets {a_{g}}_{g∈G} corresponding to graded Hecke algebras A form a vector space of dimension d + dim((⋀^{2} V)^{G}).

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

(b) ⟹: If A is a graded Hecke algebra and a_{g} ≠ 0 then by Lemma 1.8d, codim(V^{g}) = 2 and ker a_{g} = V^{g}. So a_{g} is determined by its value a_{g}(b_{1}, b_{2}) on a basis b_{1}, b_{2} of (V^{g})^{⊥}. Suppose h ∈ Z_{G}(g). Then, by Lemma 1.8e,

a_{g}(b_{1}, b_{2}) = a_{hgh^{−1}}(hb_{1}, hb_{2}) = a_{g}(hb_{1}, hb_{2}) = det(h^{⊥}) a_{g}(b_{1}, b_{2}),

and so det(h^{⊥}) = 1. Note that h(V^{g}) = V^{g} and h(V^{g})^{⊥} = (V^{g})^{⊥} since, for each v ∈ V^{g}, h(v) = hg(v) = gh(v). ⟸: If codim(V^{g}) = 2 then, up to constant multiples, there is a unique skew symmetric form on V which is nondegenerate on (V^{g})^{⊥} and which has ker a_{g} = V^{g}. Fix such a form and then define forms a_{h}, h ∈ G, by

a_{h}(v_{1}, v_{2}) = \{ \begin{matrix}a_{g}(kv_{1}, kv_{2}) & if h = k^{−1}gk, \\ 0 & otherwise, \end{matrix}  (1.10)

for v_{1}, v_{2} ∈ V. Let a_{1} be any G-invariant skew symmetric form on V. Then this collection {a_{g}}_{g∈G} 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 {a_{g}}_{g∈G}, running over all graded Hecke algebras A for G, form a vector space. Since each of the collections {a_{g}}_{g≠1} constructed by (1.10) has its support on a single conjugacy class, these collections form a basis of the vector space of sets {a_{g}}_{g≠1}. The only condition on the form a_{1} is that it satisfies (1.6), which means that it is a G-invariant element of (⋀^{2} V)^{∗}.

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

Corollary 1.11. Assume that G contains h = ξ · 1 for some ξ ∈ ℂ \ {±1}. If A is a graded Hecke algebra for G, then a_{g} = 0 for all g ≠ 1. Proof. If h = ξ · 1 ∈ G, then h ∈ Z_{G}(g) for every g ∈ G and det(h^{⊥}) = ξ^{2} if codim(V^{g}) = 2. The statement then follows from Theorem 1.9b.

2. 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 = V_{1} ⊕ V_{2} where V_{1} and V_{2} are G-invariant subspaces. The group G is a real reflection group if V = ℂ ⊗_{ℝ} V_{ℝ} for a real vector space V_{ℝ} and G ⊆ GL(V_{ℝ}).

The following facts about reflection groups are well known.

Lemma 2.1. Let G be an irreducible reflection group.

(a) [ST, Theorem 5.3] The number of elements g ∈ G such that codim(V^{g}) = 2 is ∑_{i<j} m_{i}m_{j} where m_{1}, …, m_{n} are the exponents of G.

(b) [Ca, Lemma 2] If G is a real reflection group and g ∈ G with codim(V^{g}) = 2, then g is the product of two reflections.

(c) [OT, Theorem 6.27] For any g ∈ G, the space V^{g} 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 G_{4} of rank 2, in the notation of Shephard and Todd [ST]. All the reflections have order 3 and −1 ∈ G_{4}. 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 −1 ∈ G_{4} is not a product of two reflections.

Lemma 2.2. Let G ⊆ GL(V) be a complex reflection group. Let A be a graded Hecke algebra for G and let g ∈ G. Let V^{G} = {v ∈ V | gv = v for all g ∈ G} be the invariants in V.

(a) If g = 1 and dim V^{G} ≤ 1, then a_{g} = 0.

(b) If the order of g is 2, then a_{g} = 0. Proof. (a) Let ⟨ , ⟩: V × V → ℂ be a nondegenerate G-invariant Hermitian form on V and write V = V^{G} ⊕ (V^{G})^{⊥} where (V^{G})^{⊥} = {v ∈ V | ⟨v, w⟩ = 0 for all w ∈ V^{G}}. Since dim(V^{G}) ≤ 1 and a_{1} is skew symmetric, a_{1} restricted to V^{G} is 0. There is a basis α_{1}, …, α_{k} of (V^{G})^{⊥} and constants ξ_{1}, …, ξ_{k} ∈ ℂ, ξ_{i} ≠ 1, such that the reflections s_{1}, …, s_{k} given by

s_{i}v = v + (ξ_{i} − 1)\frac{⟨v, α_{i}⟩}{⟨α_{i}, α_{i}⟩}α_{i}, for v ∈ V ,

are in G. Equation (1.6) implies that, for any v ∈ V,

a_{1}(α_{i}, v) = a_{1}(s_{i}α_{i}, s_{i}v) = a_{1} (ξ_{i}α_{i}, v + (ξ_{i} − 1)\frac{⟨v, α_{i}⟩}{⟨α_{i}, α_{i}⟩}α_{i}) = ξ_{i}a_{1}(α_{i}, v),

since a_{1}(α_{i}, α_{i}) = 0 (as a_{1} is skew symmetric). Since ξ_{i} ≠ 1, a_{1}(α_{i}, v) = 0 for 1 ≤ i ≤ k. Thus ker a_{1} = V.

(b) Since g^{2} = 1, all eigenvalues of g are ±1. If codim(V^{g}) ≠ 2, then a_{g} = 0 by Theorem 1.9b. If codim(V^{g}) = 2, then

g = id_{V^{g}} ⊕ (−id_{(V^{g})^{⊥}})

as a linear transformation on V. By [St1, Theorem 1.5], [Bou V, §5 Ex. 8], the stabilizer, Stab(V^{g}), of V^{g} is a reflection subgroup of G and so there is a reflection s ∈ Stab(V^{g}) that is the identity on V^{g}. So s ∈ Z_{G}(g) and det(s) = det(s^{⊥}) ≠ 1, where s^{⊥} is s restricted to (V^{g})^{⊥}. Thus, by Theorem 1.9b, a_{g} = 0.

2A. Real reflection groups

If G ⊆ GL(V) is a real reflection group then V = ℂ ⊗_{ℝ} V_{ℝ} and G ⊆ GL(V_{ℝ}), where V_{ℝ} 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 [Hu] or [Bou]. The action of G on V_{ℝ} has fundamental chambers wC indexed by w ∈ G. The roots for G are vectors α ∈ V_{ℝ} such that the reflections in G are the reflections s_{α} in the hyperplanes

H_{α} = {v ∈ V_{ℝ} | ⟨v, α⟩ = 0}.

For each fundamental chamber C, the reflections s_{1}, s_{2}, …, s_{n} in the hyperplanes H_{α_{1}}, H_{α_{2}}, …, H_{α_{n}} that bound C form a set of simple reflections for G. The simple reflections obtained from a different choice of fundamental chamber wC are ws_{1}w^{−1}, …, ws_{n}w^{−1}.

Theorem 2.3. Let G ⊆ GL(V_{ℝ}) be a real reflection group. Let s_{1}, …, s_{n} be a set of simple reflections in G and let m_{ij} be the order of s_{i}s_{j}. Then g ∈ G satisfies g^{2} ≠ 1, codim(V^{g}) = 2, and det(h^{⊥}) = 1 for all h ∈ Z_{G}(g) (the conditions in Theorem 1.9c) if and only if g is conjugate to

(s_{i}s_{j})^{k}, with 0 < k < \frac{m_{ij}}{2} ,

for some 1 ≤ i, j ≤ n. Proof. ⟹: Let α and β be two roots such that V^{g} = H_{α} ∩ H_{β} (see Lemma 2.1c). Then H_{α} ∩ H_{β} has nontrivial intersection with some fundamental chamber C for W, and we may assume that H_{α} and H_{β} are walls of the chamber C (since C is a cone in ℝ^{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 H_{α} and H_{β} are simple reflections and α = α_{1} and β = α_{2}.

The element g is an element of the stabilizer Stab(V^{g}), which is a reflection group by [St1, Theorem 1.5]. Since codim(V^{g}) = 2, Stab(V^{g}) is a rank two real reflection group, and therefore a dihedral group. This dihedral group is generated by the two simple reflections s_{1} and s_{2} in the hyperplanes H_{α_{1}} and H_{α_{2}} (restricted to (V^{g})^{⊥}) and all reflections have determinant −1. Let g^{⊥} be the element g restricted to (V^{g})^{⊥}. Since g ∈ Z_{G}(g), det(g^{⊥}) = 1, and so g must be a product of an even number of reflections. Thus g = (s_{1}s_{2})^{k} or g = (s_{2}s_{1})^{k}, for some 0 < k ≤ m/2, where m is the order of s_{1}s_{2}. Since g^{2} ≠ 1, k ≠ m/2, and so g is conjugate to (s_{1}s_{2})^{k} for some 0 < k < m/2. ⟸: Assume that g = (s_{i}s_{j})^{k} for some 0 < k < m_{ij}/2. Then V^{g} = H_{α_{i}} ∩ H_{α_{j}} and so codim(V^{g}) = 2. Since g is a product of an even number of reflections, det(g^{⊥}) = 1. The only elements of O(V_{ℝ}) ≅ O_{2}(ℝ) that are diagonalizable in GL(V_{ℝ}) ≅ GL_{2}(ℝ) 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_{ℝ}, only in V = ℂ ⊗_{ℝ} ℝ. Let h ∈ Z_{G}(g) and let h^{⊥} ∈ O(V_{ℝ}) ≅ O_{2}(ℝ) denote h restricted to (V^{g})^{⊥}. Since h^{⊥} commutes with g^{⊥} and g^{⊥} has distinct eigenvalues, g^{⊥} and h^{⊥} have the same eigenvectors. Hence, det h^{⊥} = 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 s_{1}, …, s_{n} using the Dynkin diagrams below. If nodes i and j and nodes j and k are connected by single edges, then s_{i}s_{j} is conjugate to s_{j}s_{k} via the element s_{i}s_{j}s_{k}.

The following table gives representatives of the conjugacy classes of g ∈ G that may have a_{g} ≠ 0 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 S_{n} acting on an ndimensional vector space V by permutation matrices, then dim(V^{G}) = 1 and, by Lemma 2.2a and

graded hecke algebras 9

Theorem 2.3, a_{g} ≠ 0 for some graded Hecke algebra A only if g is conjugate to the three cycle (1, 2, 3) = s_{1}s_{2} (this example is analyzed in Section 3).

Type  Representative g
with a_{g} ≠ 0
A_{n−1}  s_{1}s_{2}
B_{n}  s_{1}s_{2}, s_{2}s_{3}
D_{n}  s_{2}s_{3}
E_{6}, E_{7}, E_{8}  s_{1}s_{4}
F_{4}  s_{1}s_{2}, s_{2}s_{3}, s_{3}s_{4}
H_{3}, H_{4}  s_{1}s_{2}, (s_{1}s_{2})^{2}, s_{2}s_{3}
I_{2}(m)  (s_{1}s_{2})^{k}, 0 < k < m/2

Table 1. Graded Hecke algebras for real reflection groups.

A_{n−1} ∘  ∘  ∘  ∘^{1}  ^{2}  ^{n−2} ^{n−1}  ^{B_{n} ∘  ∘  ∘  ∘  ∘^{1}  ^{2}  ^{3}  ^{n−1}  ^{n}}

D_{n}  ∘ ∘ ∘ ∘ ∘ ∘  ^{n}^{n−1}^{4}^{3}^{2}^{1}........................................................................ ........................................................................  E_{6}  ∘ ∘ ∘ ∘ ∘  ∘  ^{6}^{5}^{4}^{3}^{2}  ^{1}
E_{7}  ∘ ∘ ∘ ∘ ∘ ∘  ∘  ^{7}^{6}^{5}^{4}^{3}^{2}  _{1}  E_{8}  ∘ ∘ ∘ ∘ ∘ ∘ ∘  ∘  ^{8}^{7}^{6}^{5}^{4}^{3}  _{1}^{2}

F_{4}  ∘  ∘  ∘  ∘^{4}^{3}^{2}^{1}  ^{H_{3}  ∘  ∘  ∘^{3}^{2}^{5}^{1}}
H_{4}  ∘  ∘  ∘  ∘^{4}^{3}^{2}^{5}^{1}  ^{I_{2}(m) ∘  ∘^{1} ^{m} ^{2}}

Figure 1. Coxeter-Dynkin diagrams for real reflection groups.

2B. Complex reflection groups

The irreducible complex reflection groups were classified by Shephard and Todd [ST]. There is one infinite family denoted G(r, p, n) and a list of exceptional complex reflection groups denoted G_{4}, …, G_{35}. 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 = e^{2πi/r}. Let S_{n} be the symmetric group of n × n matrices and let

ξ_{j} = diag(1, 1, …, 1, ξ, 1, …, 1),

where ξ appears in the jth entry. Then

G(r, p, n) = {ξ_{1}^{λ_{1}} ⋯ ξ_{n}^{λ_{n}}w | w ∈ S_{n}, 0 ≤ λ_{i} ≤ r − 1, λ_{1} + ⋯ + λ_{n} = 0 mod p}.

For λ = (λ_{1}, …, λ_{n}) ∈ (ℤ/rℤ)^{n}, let ξ^{λ} = ξ_{1}^{λ_{1}} ⋯ ξ_{n}^{λ_{n}}. Then the multiplication in G(r, p, n) is described by the relations

ξ^{λ}ξ^{μ} = ξ^{λ+μ}  and  wξ^{λ} = ξ^{wλ}w,  for λ, μ ∈ (ℤ/rℤ)^{n}, w ∈ S_{n},

where S_{n} acts on (ℤ/rℤ)^{n} by permuting the factors. Let v_{i} be the column vector with 1 in the i^{th} entry and all other entries 0. The group G(r, p, n) acts on V := ℂ^{n} with orthonormal basis {v_{1}, …, v_{n}} 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 S_{n},

(b) G(2, 1, n) is the Weyl group W B_{n} of type B_{n},

(c) G(2, 2, n) is the Weyl group W D_{n} of type D_{n}, and

(d) G(r, r, 2) is the dihedral group I_{2}(r) of order 2r.

The reflections in G(r, p, n) are

\begin{matrix}ξ_{i}^{kp}, & 1 ≤ i ≤ n, 0 ≤ k ≤ (r/p) − 1, and \\ ξ_{i}^{k}ξ_{j}^{−k}(i, j), & 1 ≤ i < j ≤ n, 0 ≤ k ≤ r − 1, \end{matrix}

where (i, j) is the transposition in S_{n} that switches i and j. Conjugacy in G(r, p, n). Each element of G(r, p, n) is conjugate by elements of S_{n} to a disjoint product of cycles of the form

ξ_{i}^{λ_{i}} ⋯ ξ_{k}^{λ_{k}}(i, i + 1, …, k).

By conjugating this cycle by ξ_{i}^{−c}ξ_{i+1}^{λ_{i}}ξ_{i+2}^{λ_{i}+λ_{i+1}} ⋯ ξ_{k}^{λ_{i}+···+λ_{k−1}} ∈ G(r, r, n), we have

ξ_{i}^{−c}ξ_{k}^{c+λ_{i}+···+λ_{k}}(i, …, k), where c = (k − i)λ_{i} + (k − i − 1)λ_{i+1} + ⋯ + λ_{k−1}.

If i_{1}, i_{2}, …, i_{ℓ} denote the minimal indices of the cycles and c_{1}, …, c_{ℓ} are the numbers c for the various cycles, then after conjugating by ξ_{i_{1}}^{c_{1}} ⋯ ξ_{i_{ℓ−1}}^{c_{ℓ−1}}ξ_{i_{ℓ}}^{−(c_{1}+···+c_{ℓ−1})} ∈ G(r, r, n), each cycle becomes

ξ_{k}^{λ_{i}+···+λ_{k}}(i, …, k)  except the last, which is  ξ_{i_{ℓ}}^{−a}ξ_{n}^{b}(i_{ℓ}, …, n),

where a = c_{1} + ⋯ + c_{ℓ} and b = a + λ_{i_{ℓ}} + ⋯ + λ_{n}. If k = n − i_{ℓ} + 1 is the length of the last cycle, then conjugating the last cycle by ξ_{i_{ℓ}}^{k−1}ξ_{i_{ℓ}+1}^{−1} ⋯ ξ_{n}^{−1} ∈ G(r, r, n) gives

ξ_{i_{ℓ}}^{−a+k}ξ_{n}^{b−k}(i_{ℓ}, …, n).

If we conjugate the last cycle by ξ_{i_{ℓ}}^{p} ∈ G(r, p, n), we have

ξ_{i_{ℓ}}^{−a+p}ξ_{n}^{b−p}(i_{ℓ}, …, 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

ξ_{k}^{a}(i, i + 1, …, k),  0 ≤ a ≤ r − 1,  (2.4a)

except possibly the last cycle, which is of the form

ξ_{i_{ℓ}}^{a}ξ_{n}^{b}(i_{ℓ}, i_{ℓ} + 1, …, n),  with 0 ≤ a ≤ gcd(p, k) − 1,  (2.4b)

where k = n − i_{ℓ} + 1 is the length of the last cycle.

Centralizers in G(r, p, n). Let Z_{G(r,p,n)}(g) = {h ∈ G(r, p, n) | hg = gh} denote the centralizer of g ∈ G(r, p, n). Since G(r, p, n) is a subgroup of G(r, 1, n),

Z_{G(r,p,n)}(g) = Z_{G(r,1,n)}(g) ∩ G(r, p, n),

for any element g ∈ G(r, p, n). Suppose that g is an element of G(r, 1, n) which is a product of disjoint cycles of the form ξ_{k}^{a}(i, …, k) and that h ∈ G(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, ξ_{k}^{a}(i, …, k) and ξ_{m}^{b}(j, …, m) with b = a and k −i = m−j,

(b) conjugating a single cycle ξ_{k}^{a}(i, …, k) by powers of itself, and

(c) conjugating a single cycle ξ_{k}^{a}(i, …, k) by ξ_{i}^{b} ⋯ ξ_{k}^{b}, for any 0 ≤ b ≤ r − 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 [Mac, p. 170]) of the number of such operations shows that if g ∈ G(r, 1, n) and m_{a,k} is the number of cycles of type ξ_{i+k}^{a}(i, i + 1, …, i + k) for g, then

Card(Z_{G(r,1,n)}(g)) = ∏_{a,k}(m_{a,k}! · k^{m_{a,k}} · r^{m_{a,k}}) .  (2.5)

Determining the graded Hecke algebras for G(r, p, n). It follows from Lemma 1.8a that if g = ξ_{i}^{a+b}ξ_{k}^{−a}(i, …, k), then (V^{g})^{⊥} has basis

{v_{k} − v_{k−1}, v_{k−1} − v_{k−2}, …, v_{i+1} − ξ^{a}v_{i}} if b = 0,  and  {v_{i}, …, v_{k}} if b ≠ 0.

Thus, if g ∈ G(r, p, n) and codim(V^{g}) = 2, then g is conjugate to one of the following elements:

b = ξ_{1}^{a}ξ_{3}^{−a}(1, 2, 3),  0 ≤ a ≤ gcd(p, 3) − 1,
c = ξ_{1}^{a+ℓ}ξ_{2}^{−a}(1, 2),  ℓ ≠ 0 (so r ≠ 1),
d = ξ_{1}^{ℓ_{1}}ξ_{2}^{ℓ_{2}},  ℓ_{1} ≠ 0, ℓ_{2} ≠ 0 (so r ≠ 1),
e = (1, 2)ξ_{3}^{ℓ},  ℓ ≠ 0,
f = (1, 2)ξ_{3}^{a}ξ_{4}^{−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)) = {ξ_{1}^{ℓ} ⋯ ξ_{n}^{ℓ} | nℓ = 0 mod p}.

Since ξ_{1}^{p} ⋯ ξ_{n}^{p} ∈ Z(G(r, p, n)), it follows that p = r or p = r/2 whenever Z(G(r, p, n)) ⊆ {±1} = {ξ_{1}^{0} ⋯ ξ_{n}^{0}, ξ_{1}^{r/2} ⋯ ξ_{n}^{r/2}}.

(b1) If n ≥ 4, the element ξ_{1}ξ_{2}ξ_{3}ξ_{4}^{−3} ∈ Z_{G}(b) and has determinant ξ^{2} on (V^{b})^{⊥} = span-{v_{3} − v_{2}, v_{2} − ξ^{a}v_{1}}.

(b2) If n = 3 and p = 0 mod 3, the element ξ_{1}^{p/3}ξ_{2}^{p/3}ξ_{3}^{p/3} ∈ Z_{G}(b) and has determinant ξ^{2p/3} on (V^{b})^{⊥}.

(c1) If n ≥ 3, the element ξ_{1}ξ_{2}ξ_{3}^{−2} ∈ Z_{G}(c) and has determinant ξ^{2} on (V^{c})^{⊥} = span-{v_{1}, v_{2}}.

(c2) If n = 2, p = r/2 and p is odd, the element ξ_{1}^{p/4}ξ_{2}^{p/4} ∈ Z_{G}(c) and has determinant ξ^{r/2} on (V^{c})^{⊥}.

(d1) If n ≥ 3, the element ξ_{1}ξ_{3}^{−1} ∈ Z_{G}(d) and has determinant ξ on (V^{d})^{⊥} = span-{v_{1}, v_{2}}.

(d2) If p = r/2, the element ξ_{1}^{r/2} ∈ Z_{G}(d) and has determinant ξ^{r/2} on (V^{d})^{⊥}.

(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

a_{b} = 0  unless  (i) r = 1, or

(ii) r = 2, or

(iii) n = 3 and p ≠ 0 mod 3,

a_{c} = 0  unless  (i) r = 2 and p = 1, or

(ii) n = 2 and p = r/2,

a_{d} = 0  unless  p = r, n = 2 and p ≠ 0 mod 2,
a_{e} = 0  always,  and
a_{f} = 0  always.

In the remaining cases, one uses the description of Z_{G}(g) given just before (2.5) to check that all elements of Z_{G}(g) have determinant 1 on (V^{g})^{⊥}. Note that n = 3 and p ≠ 0 mod 3 imply that a_{b} = 0 for the elements b = ξ_{1}^{a}ξ_{3}^{−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) ⊗ ℂG always exists and corresponds to the case when all of the skew symmetric forms a_{g} are zero). The table below gives representatives of the conjugacy classes of g ∈ G that may have a_{g} ≠ 0 for some graded Hecke algebra A.

Group  Representative g
with a_{g} ≠ 0
G(1, 1, n) = S_{n}  (1, 2, 3)
G(2, 1, n) = W B_{n}, n ≥ 3  ξ_{1}(1, 2), (1, 2, 3)
G(2, 2, n) = W D_{n}, n ≥ 3  (1, 2, 3)
G(r, r, 2) = I_{2}(r)  ξ_{1}^{k}ξ_{2}^{r−k}, 0 < k < r/2
G(r, r/2, 2), r/2 odd  ξ_{2}^{r/2}(1, 2)
G(r, r, 3), r ≠ 0 mod 3  (1, 2, 3)
G(r, r/2, 3), r/2 ≠ 0 mod 3, r ≠ 2  (1, 2, 3)

Table 2. Graded Hecke algebras for the groups G(r, p, n).

2C. Exceptional complex reflection groups

The irreducible exceptional complex reflection groups G are denoted G_{4}, …, G_{35} in the classification of Shephard and Todd. From Table VII in [ST], one sees that the center of G is ±1 only in the cases G_{4}, G_{12}, G_{24} and G_{33}. 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 a_{g} ≠ 0) are G_{4}, G_{12}, G_{24} and G_{33} (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 G_{4} has order 24 and seven conjugacy classes. The following data concerning these conjugacy classes are obtained from the computer software GAP [S+] using the package CHEVIE [G+]. In the following table, ω is a primitive cube root of unity and C(g) denotes the conjugacy class of g.

Conjugacy class representatives for G_{4}

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
|Z_{G}(g)|  24  4  6  6  6  6  24

The elements with determinant 1 and order more than 2 in G_{4} all have order 4. If g is an element of order 4, then |Z_{G}(g)| = 4 and every element of Z_{G}(g) has determinant 1 since Z_{G}(g) is generated by g. Hence, by Theorem 1.9b and Lemma 2.2, a_{g} can be nonzero for a graded Hecke algebra for G_{4} exactly when g has order 4. Thus, the dimension of the space of parameters for graded Hecke algebras of G_{4} is 1.

The rank 2 group G_{12} has order 48. The computer software GAP provides the following information about the conjugacy classes of G_{12}.

Conjugacy class representatives for G_{12}

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
|Z_{G}(g)|  48  4  8  6  8  48  6  8

If g is an element in G_{12} 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 h^{2} of order 4. Hence, by Theorem 1.9b, if g has order 4, then a_{g} = 0. Let g_{6} be a representative from the class of elements of order 6. Since |Z_{G}(g_{6})| = 6, Z_{G}(g_{6}) is generated by g_{6} and hence every element of Z_{G}(g_{6}) has determinant 1. We can choose g_{6}^{2} as a representative for the conjugacy class of elements of order 3. As g_{6} and g_{6}^{2} commute, ⟨g_{6}⟩ ⊂ Z_{G}(g_{6}^{2}). But |⟨g_{6}⟩| = 6 = |Z_{G}(g_{3})|, so Z_{G}(g_{6}^{2}) is generated by g_{6} and every element of Z_{G}(g_{3}) has determinant 1. Thus, a_{g} can be nonzero for a graded Hecke algebra A for G_{12} exactly when g has order 3 or 6. Thus, the dimension of the space of parameters of graded Hecke algebras for G_{12} is 2.

The rank 3 group G_{24} has order 336. Note that −1 ∈ G_{24} since Z(G) = {±1}. Up to G-orbits, there are two codimension 2 subspaces, L and M, that are equal to V^{g} for some g ∈ G_{24} (see [OT, App. C, Table C.5]). Furthermore, Stab(L) ≅ A_{2} and Stab(M) ≅ B_{2}. We need only consider elements of order 3 in Stab(L) ≅ A_{2} and of order 4 in Stab(M) ≅ B_{2} (as the rest have order 1 or 2). In G_{24}, 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.

Certain classes of G_{24}

Order(g)  3  4
det(g)  1  1
|C(g)|  56  42
|Z_{G}(g)|  6  8

If g has order 3, Z_{G}(g) must contain 1, g, g^{2}, and −1, and hence Z_{G}(g) is generated by these elements since |Z_{G}(g)| = 6. Thus all elements of Z_{G}(g) have determinant 1 on (V^{g})^{⊥}. If g has order 4 and determinant 1, then Z_{G}(g) must contain 1, g, g^{2}, g^{3}, and −1, elements which all have determinant 1 on (V^{g})^{⊥}. Since |Z_{G}(g)| = 8, these elements generate Z_{G}(g) and so every element of Z_{G}(g) has determinant 1 on (V^{g})^{⊥}. Hence, a_{g} can be nonzero for a graded Hecke algebra of G_{24} 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 G_{24} is 2.

The group G_{33} 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 V^{g} for some g ∈ G_{33} (see [OT, App. C, Table C.14]). Furthermore, Stab(L) ≅ A_{1} × A_{1} and Stab(M) ≅ A_{2}. We need not consider the case where V^{g} = L since then g has order 2 and hence a_{g} = 0 for any graded Hecke algebra by Proposition 2.2b.

We use a presentation for G_{33} in six coordinates from [ST]: Let V := {(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}) | x_{i} ∈ ℂ} and consider the group generated by order 2 reflections about the hyperplanes H_{1} = {x_{2} −x_{3} = 0}, H_{2} = {x_{3} − x_{4} = 0}, H_{3} = {x_{1} − x_{2} = 0}, H_{4} = {x_{1} − ωx_{2} = 0}, H_{5} = {x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} = 0}, where ω is a primitive cube root of unity. The fixed point space of this (reducible) group is Y = H_{1} ∩ ⋯ ∩ H_{5} = {(0, 0, 0, 0, x, −x) | x ∈ ℂ}, and G_{33} is just the restriction to Y^{⊥}. Let s_{i} be the order 2 reflection about H_{i}. Let g = s_{1}s_{3}. Then V^{g} = H_{1} ∩ H_{3} and Stab(V^{g}) ≅ A_{2}. Let h = (s_{1}s_{3}s_{4})^{2}, the diagonal matrix with diagonal {ω, ω, ω, 1, 1, 1}. Then h acts as ω times the identity on (V^{g})^{⊥} as (V^{g})^{⊥} ⊆ ℂ-span{x_{1}, x_{2}, x_{3}}. Hence, h commutes with g. But (V^{g})^{⊥} has dimension 2 and h has determinant ω^{2} ≠ 1 on (V^{g})^{⊥}. Thus, by Theorem 1.9b and Lemma 2.2, a_{g} = 0 for any graded Hecke algebra. The same argument applied to Y^{⊥} shows that G_{33} has no nontrivial graded Hecke algebras. In summary, the dimension of the space of parameters for graded Hecke algebras for G_{33} is zero.

Group  g with a_{g} ≠ 0
G_{4}  Order(g) = 4
G_{12}  Order(g_{1}) = 3 and Order(g_{2}) = 6
G_{24}  Order(g_{1}) = 3 and Order(g_{2}) = 4, det(g_{2}) = 1

Table 3. Graded Hecke algebras for exceptional complex reflection groups.

3. The graded Hecke algebras H_{gr}

In [Lu], 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 α_{1}, …, α_{n} be a choice of simple roots in R and let s_{1}, …, s_{n} be the corresponding simple reflections in W. Let s_{α} be the reflection in the root α so that, for v ∈ V,

s_{α}v = v − ⟨v, α^{∨}⟩α, where α^{∨} = 2α/⟨α, α⟩.

Let R^{+} = {α > 0} denote the set of positive roots in R.

Let k_{α} be fixed complex numbers indexed by the roots α ∈ R satisfying

k_{wα} = k_{α},  for all w ∈ W, α ∈ R.  (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 ℂW = ℂ-span{t_{g} | g ∈ W }, with t_{g}t_{h} = t_{gh}, and let S(V) be the symmetric algebra of V. Lusztig [Lu] defines the “graded Hecke algebra” with parameters {k_{α}} to be the unique algebra structure H_{gr} on the vector space S(V) ⊗ ℂW such that (3.2 a) S(V) = S(V) ⊗ 1 is a subalgebra of H_{gr}, (3.2 b) ℂW = 1 ⊗ ℂW is a subalgebra of H_{gr}, and (3.2 c) t_{s_{i}}v = (s_{i}v)t_{s_{i}} − k_{α_{i}}⟨v, α_{i}^{∨}⟩, for all v ∈ V and simple reflections s_{i} in the simple roots α_{i}.

We shall show that every algebra H_{gr} as defined by (3.2 a-c) is a graded Hecke algebra A for a specific set of skew symmetric bilinear forms a_{g}.

Let k_{α} ∈ ℂ as in (3.1). Use the notation

h = \frac{1}{2} ∑_{α>0} k_{α}α^{∨}t_{s_{α}},  so that  ⟨v, h⟩ = \frac{1}{2} ∑_{α>0} k_{α}⟨v, α^{∨}⟩t_{s_{α}}  (3.3)

for v ∈ V. The element h should be viewed as an element of V ⊗ ℂW, and ⟨v, h⟩ ∈ ℂW. With this notation, let A be the algebra (as in Section 1) generated by V and ℂW with relations

t_{g}v = (gv)t_{g}  and  [v, w] = −[⟨v, h⟩, ⟨w, h⟩],  for v, w ∈ V, g ∈ W.  (3.4)

Note that A is defined by the bilinear forms

a_{g}(v, w) = \frac{1}{4} ∑_{{α, β > 0 \atop g = s_{α}s_{β}}} k_{α}k_{β} (⟨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 H_{gr}.

Theorem 3.5. Let W be a finite real reflection group and let A be the algebra defined by (3.4).

(a) As vector spaces, A ≅ S(V) ⊗ ℂW (and hence, A is a graded Hecke algebra).

(b) If ṽ = v − ⟨v, h⟩ for v ∈ V, then

[ṽ, w̃] = 0  and  t_{s_{i}} ṽ = (s̃_{i}v)t_{s_{i}} − k_{α_{i}}⟨v, α_{i}^{∨}⟩,

for all v, w ∈ V and simple reflections s_{i} in W. Proof. First note that if u, v ∈ V then

[u, ⟨v, h⟩] = \frac{1}{2} ∑_{α>0} k_{α}⟨v, α^{∨}⟩⟨u, α^{∨}⟩αt_{s_{α}} = [v, ⟨u, h⟩].  (∗)

Thus, for u, v, w ∈ V,

\begin{matrix}[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. \end{matrix}  (3.6)

For v ∈ V, h ∈ W, and s_{i} a simple reflection,

\begin{aligned}t_{s_{i}}⟨v, h⟩t_{s_{i}} & = \frac{1}{2} ∑_{α>0} k_{α}⟨v, α^{∨}⟩t_{s_{s_{i}α}} = (\frac{1}{2} ∑_{α>0} k_{α}⟨v, s_{i}α^{∨}⟩t_{s_{α}}) + k_{α_{i}}⟨v, α_{i}^{∨}⟩t_{s_{i}} \\ & = (\frac{1}{2} ∑_{α>0} k_{α}⟨s_{i}v, α^{∨}⟩t_{s_{α}}) + k_{α_{i}}⟨v, α_{i}^{∨}⟩t_{s_{i}} = ⟨s_{i}v, h⟩ + k_{α_{i}}⟨v, α_{i}^{∨}⟩t_{s_{i}}. \end{aligned}  (3.7)

Using this equality, we obtain

\begin{aligned}t_{s_{i}}[v, w]t_{s_{i}} & = −t_{s_{i}}[⟨v, h⟩, ⟨w, h⟩]t_{s_{i}} \\ & = −[⟨s_{i}v, h⟩ + k_{α_{i}}⟨v, α_{i}^{∨}⟩t_{s_{i}}, ⟨s_{i}w, h⟩ + k_{α_{i}}⟨w, α_{i}^{∨}⟩t_{s_{i}}] \\ & = [s_{i}v, s_{i}w] − k_{α_{i}}⟨v, α_{i}^{∨}⟩[t_{s_{i}}, ⟨s_{i}w, h⟩] − k_{α_{i}}⟨w, α_{i}^{∨}⟩[⟨s_{i}v, h⟩, t_{s_{i}}] \\ & = [s_{i}v, s_{i}w] − k_{α_{i}}⟨v, α_{i}^{∨}⟩(t_{s_{i}}⟨s_{i}w, h⟩t_{s_{i}} − ⟨s_{i}w, h⟩)t_{s_{i}} \\ & + k_{α_{i}}⟨w, α_{i}^{∨}⟩(t_{s_{i}}⟨s_{i}v, h⟩t_{s_{i}} − ⟨s_{i}v, h⟩)t_{s_{i}} \\ & = [s_{i}v, s_{i}w] − k_{α_{i}}⟨v, α_{i}^{∨}⟩(⟨w, h⟩t_{s_{i}} + k_{α_{i}}⟨s_{i}w, α_{i}^{∨}⟩ − ⟨s_{i}w, h⟩t_{s_{i}}) \\ & + k_{α_{i}}⟨w, α_{i}^{∨}⟩(⟨v, h⟩t_{s_{i}} + k_{α_{i}}⟨s_{i}v, α_{i}^{∨}⟩ − ⟨s_{i}v, h⟩t_{s_{i}}) \\ & = [s_{i}v, s_{i}w] − k_{α_{i}}⟨v, α_{i}^{∨}⟩⟨w, α_{i}^{∨}⟩⟨α_{i}, h⟩t_{s_{i}} − k_{α_{i}}^{2}⟨v, α_{i}^{∨}⟩⟨w, s_{i}α_{i}^{∨}⟩ \\ & + k_{α_{i}}⟨w, α_{i}^{∨}⟩⟨v, α_{i}^{∨}⟩⟨α_{i}, h⟩t_{s_{i}} + k_{α_{i}}^{2}⟨w, α_{i}^{∨}⟩⟨v, s_{i}α_{i}^{∨}⟩ \\ & = [s_{i}v, s_{i}w]. \end{aligned}  (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) ⊗ ℂW.

(b) This can now be proved by direct computation. If v, w ∈ V then

[ṽ, w̃] = [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 v ∈ V and s_{i} is a simple reflection then, by (3.7),

t_{s_{i}} ṽt_{s_{i}} = t_{s_{i}}vt_{s_{i}} − t_{s_{i}}⟨v, h⟩t_{s_{i}} = s_{i}v − ⟨s_{i}v, h⟩ − k_{α_{i}}⟨v, α_{i}^{∨}⟩t_{s_{i}} = s̃_{i}v − k_{α_{i}}⟨v, α_{i}^{∨}⟩t_{s_{i}}.

Theorem 3.5b shows that if A is the graded Hecke algebra defined by (3.4), then the elements ṽ, for v ∈ V, generate a subalgebra of A isomorphic to S(V) and these elements together with the t_{s_{i}} satisfy the relations of (3.2 c). Since part (a) of Theorem 3.5 shows that A is isomorphic to S(V) ⊗ ℂW 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 F_{4}, H_{3}, H_{4}, and I_{2}(m).

4. Examples

4A. The symmetric group G(1, 1, n) = S_{n}

Let V be an n dimensional vector space with orthonormal basis v_{1}, …, v_{n} and let S_{n} act on V by permuting the v_{i}. Let A be a graded Hecke algebra for S_{n}. 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,

[v_{i}, v_{j}] = ∑_{k≠i,j}(a_{(i,j,k)}(v_{i}, v_{j})t_{(i,j,k)} + a_{(j,i,k)}(v_{i}, v_{j})t_{(j,i,k)}),

since v_{i} or v_{j} is in V^{g} = ker a_{g} for all other three cycles g. Since, by (1.6), a_{(j,i,k)}(v_{i}, v_{j}) = a_{(i,j,k)}(v_{j}, v_{i}) = −a_{(i,j,k)}(v_{i}, v_{j}), the graded Hecke algebra A is defined by the relations

[v_{i}, v_{j}] = β ∑_{k≠i,j}(t_{(i,j,k)} − t_{(j,i,k)})  and t_{w}v_{i} = v_{w(i)}t_{w},  (4.1)

where w ∈ S_{n}, 1 ≤ i, j ≤ n, i ≠ j, and β = a_{(1,2,3)}(v_{1}, v_{2}).

Let k ∈ ℂ. Then, with h as in (3.3),

⟨v_{i}, h⟩ = \frac{1}{2} ∑_{ℓ<m} k⟨v_{i}, v_{ℓ} − v_{m}⟩t_{(ℓ,m)} = \frac{k}{2} (∑_{i<ℓ} t_{(i,ℓ)} − ∑_{i>ℓ} t_{(ℓ,i)}) = \frac{k}{2} ∑_{i≠ℓ} sgn(ℓ − i)t_{(i,ℓ)}. (4.2)

If f ∈ ℂS_{n}, let f|_{t_{g}} denote the coefficient of t_{g} in f. Let A be the graded Hecke algebra defined by the relations in (4.1) with

\begin{aligned}β & = a_{(i,j,ℓ)}(v_{i}, v_{j}) = [⟨v_{i}, h⟩, ⟨v_{j}, h⟩]|_{t_{(i,j,ℓ)}} \\ & = (k^{2}/4)(t_{(i,ℓ)}t_{(j,ℓ)} + t_{(i,j)}t_{(i,ℓ)} − t_{(j,ℓ)}t_{(i,j)})|_{t_{(i,j,ℓ)}} = k^{2}/4. \end{aligned}  (4.3)

If ṽ_{i} = v_{i} − ⟨v_{i}, h⟩ and s_{i} is the simple reflection (i, i + 1) then, by Theorem 3.5,

\begin{aligned}ṽ_{i}ṽ_{j} & = ṽ_{j}ṽ_{i},  t_{s_{i}} ṽ_{i} = ṽ_{i+1}t_{s_{i}} + k,  t_{s_{i}} ṽ_{i+1} = ṽ_{i}t_{s_{i}} − k, and \\ t_{s_{j}} ṽ_{i} & = ṽ_{i}t_{s_{j}},  for |i − j| > 1, \end{aligned}  (4.4)

and the algebra A is the graded Hecke algebra H_{gr} for S_{n} which is defined in Section 3. When k = 1, the map

\begin{matrix}A & ⟶ & ℂS_{n} \\ t_{w} & ⟼ & t_{w} \\ v_{i} & ⟼ & \frac{1}{2} ∑_{ℓ≠i} t_{(i,ℓ)} \end{matrix}  (4.5)

is a surjective algebra homomorphism.

4B. The hyperoctahedral group G(2, 1, n) = W B_{n}

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 {v_{1}, …, v_{n}}. In this case, ξ_{i} 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 β_{1} = a_{(i,j,k)}(v_{i}, v_{j}) and β_{2} = a_{ξ_{1}(1,2)}(v_{1}, v_{2}), then, in the algebra A,

[v_{i}, v_{j}] = β_{2}(t_{ξ_{1}(1,2)} − t_{ξ_{2}(1,2)}) + β_{1} ∑_{ℓ≠i,j} ( \begin{matrix}t_{(i,j,ℓ)} − t_{ξ_{i}ξ_{ℓ}(i,j,ℓ)} − t_{ξ_{i}ξ_{j}(i,j,ℓ)} + t_{ξ_{j}ξ_{ℓ}(i,j,ℓ)} \\ +t_{ξ_{i}ξ_{j}(j,i,ℓ)} + t_{ξ_{j}ξ_{ℓ}(j,i,ℓ)} − t_{ξ_{i}ξ_{ℓ}(j,i,ℓ)} − t_{(j,i,ℓ)} \end{matrix} ) . (4.6)

Let k_{s}, k_{ℓ} ∈ ℂ. Then, with h as in (3.3),

\begin{aligned}⟨v_{i}, h⟩ & = \frac{k_{s}}{2} ∑_{ℓ}⟨v_{i}, 2v_{ℓ}⟩t_{ξ_{ℓ}} − \frac{k_{ℓ}}{2} ∑_{ℓ<m}⟨v_{i}, v_{ℓ} − v_{m}⟩t_{(ℓ,m)} − \frac{k_{ℓ}}{2} ∑_{ℓ<m}⟨v_{i}, v_{ℓ} + v_{m}⟩t_{ξ_{ℓ}(ℓ,m)} \\ & = k_{s}t_{ξ_{i}} − \frac{k_{ℓ}}{2} (∑_{i<ℓ}(t_{(i,ℓ)} + t_{ξ_{i}ξ_{ℓ}(i,ℓ)}) + ∑_{i>ℓ}(−t_{(i,ℓ)} + t_{ξ_{i}ξ_{ℓ}(i,ℓ)})) . \end{aligned}  (4.7)

If f ∈ ℂG(2, 1, n), let f|_{t_{g}} denote the coefficient of t_{g} in f. With notation as in (4.6), let A be the graded Hecke algebra for G(2, 1, n) with

β_{1} = a_{(i,j,ℓ)}(v_{i}, v_{j}) = [⟨v_{i}, h⟩, ⟨v_{j}, h⟩]|_{t_{(i,j,ℓ)}}
= (k_{ℓ}^{2}/4)(t_{(i,ℓ)}t_{(j,ℓ)} + t_{(i,j)}t_{(i,ℓ)} − t_{(j,ℓ)}t_{(i,j)})|_{t_{(i,j,ℓ)}} = k_{ℓ}^{2}/4, and
β_{2} = [⟨v_{i}, h⟩, ⟨v_{j}, h⟩]|_{t_{ξ_{i}(i,j)}}
= (1/2)k_{s}k_{ℓ}(−t_{ξ_{i}}t_{(i,j)} + t_{(i,j)}t_{ξ_{j}} − t_{ξ_{j}}t_{ξ_{i}ξ_{j}(i,j)} − t_{ξ_{i}ξ_{j}(i,j)}t_{ξ_{i}})|_{t_{ξ_{i}(i,j)}} = −k_{s}k_{ℓ}.

If ṽ_{i} = v_{i} − ⟨v_{i}, h⟩, then, by Theorem 3.5, the ṽ_{i} commute and the algebra A is the algebra H_{gr} for W B_{n} defined in Section 3.

4C. The type D_{n} Weyl group G(2, 2, n) = W D_{n}

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 {v_{1}, …, v_{n}}. This is an index 2 subgroup of G(2, 1, n), and our notation is the same as used above for W B_{n}.

Let A be a graded Hecke algebra for G(2, 2, n). If β = a_{(i,j,k)}(v_{i}, v_{j}) then, in the algebra A,

[v_{i}, v_{j}] = β ∑_{ℓ≠i,j} ( \begin{matrix}t_{(i,j,k)} − t_{ξ_{i}ξ_{ℓ}(i,j,ℓ)} − t_{ξ_{i}ξ_{j}(i,j,ℓ)} + t_{ξ_{j}ξ_{ℓ}(i,j,ℓ)} \\ +t_{ξ_{i}ξ_{j}(j,i,ℓ)} + t_{ξ_{j}ξ_{ℓ}(j,i,ℓ)} − t_{ξ_{i}ξ_{ℓ}(j,i,ℓ)} − t_{(j,i,ℓ)} \end{matrix} ) .  (4.8)

Let k ∈ ℂ. Then, with h as in (3.3),

⟨v_{i}, h⟩ = \frac{k}{2} (∑_{i<ℓ}(t_{(i,ℓ)} + t_{ξ_{i}ξ_{ℓ}(i,ℓ)}) + ∑_{i>ℓ}(−t_{(i,ℓ)} + t_{ξ_{i}ξ_{ℓ}(i,ℓ)})) .  (4.9)

If f ∈ ℂG(2, 2, n), let f|_{t_{g}} denote the coefficient of t_{g} in f. With notation as in (4.8), let A be the graded Hecke algebra for G(2, 2, n) with

\begin{aligned}β & = a_{(i,j,ℓ)}(v_{i}, v_{j}) = [⟨v_{i}, h⟩, ⟨v_{j}, h⟩]|_{t_{(i,j,ℓ)}} \\ & = (k^{2}/4)(t_{(i,ℓ)}t_{(j,ℓ)} + t_{(i,j)}t_{(i,ℓ)} − t_{(j,ℓ)}t_{(i,j)})|_{t_{(i,j,ℓ)}} = k^{2}/4. \end{aligned}

If ṽ_{i} = v_{i} − ⟨v_{i}, h⟩, then, by Theorem 3.5, the ṽ_{i} commute and the algebra A is the algebra H_{gr} for W D_{n} defined in Section 3.

4D. The dihedral group I_{2}(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 {v_{1}, v_{2}}. The group G(r, r, 2) is realized as a real reflection group by using the basis

ε_{1} = \frac{1}{\sqrt{2}}(v_{1} + v_{2}), ε_{2} = \frac{−1}{i\sqrt{2}}(v_{1} − v_{2}).

This basis is also orthonormal and, with respect to this basis, G(r, r, 2) acts by the matrices

( \begin{matrix}cos(2πm/r) & ∓ sin(2πm/r) \\ sin(2πm/r) & ± cos(2πm/r) \end{matrix} ) , 0 ≤ m ≤ r − 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 {ξ_{1}^{k}ξ_{2}^{−k}, ξ_{1}^{−k}ξ_{2}^{k}}, 0 < k < r/2. Then, in the algebra A,

[ε_{1}, ε_{2}] = ∑_{0<k<r/2} β_{k}(t_{ξ_{1}^{k}ξ_{2}^{−k}} − t_{ξ_{1}^{−k}ξ_{2}^{k}}),  where β_{k} = a_{ξ_{1}^{k}ξ_{2}^{−k}}(ε_{1}, ε_{2}).  (4.10)

When r is even, there are two conjugacy classes of reflections

{ξ_{1}^{2k}ξ_{2}^{−2k}(1, 2) | 0 ≤ k < r/2}  and  {ξ_{1}^{2k+1}ξ_{2}^{−(2k+1)}(1, 2) | 0 ≤ k < r/2}.

The reflection ξ_{1}^{m}ξ_{2}^{−m}(12) is the reflection in the line perpendicular to the vector

α_{m} = sin(−2πm/2r)ε_{1} + cos(−2πm/2r)ε_{2},

and the vectors α_{m} can be taken as a root system for G(r, r, 2). With h as in (3.3) and k_{s}, k_{ℓ} ∈ ℂ,

⟨ε_{1}, h⟩ = ∑_{0≤k<r/2} (k_{s} sin(−2k 2π/2r)t_{ξ_{1}^{2k}ξ_{2}^{−2k}(1,2)} + k_{ℓ} sin(−(2k + 1) 2π/2r)t_{ξ_{1}^{2k+1}ξ_{2}^{−(2k+1)}(1,2)}) ,

⟨ε_{2}, h⟩ = ∑_{0≤k<r/2} (k_{s} cos(−2k 2π/2r)t_{ξ_{1}^{2k}ξ_{2}^{−2k}(1,2)} + k_{ℓ} cos(−(2k + 1) 2π/2r)t_{ξ_{1}^{2k+1}ξ_{2}^{−(2k+1)}(1,2)}) .

(4.11)

If f ∈ ℂG(r, r, 2), let f|_{t_{g}} denote the coefficient of t_{g} in f. With notation as in 4.10, let A be the graded Hecke algebra for G(r, r, 2) with

β_{k} = a_{ξ_{1}^{k}ξ_{2}^{−k}}(ε_{1}, ε_{2}) = [⟨ε_{1}, h⟩, ⟨ε_{2}, h⟩]|_{t_{ξ^{k}_{1}ξ{−k \atop 2}}} = \{ \begin{matrix}sin(k 2π/2r)rk_{s}k_{ℓ} & if k is odd \\ sin(k 2π/2r)\frac{r}{2}(k_{s}^{2} + k_{ℓ}^{2}) & if k is even. \end{matrix} (4.12)

If ε̃_{i} = ε_{i} − ⟨ε_{i}, h⟩, then by Theorem 3.5, the ε̃_{i} commute and the algebra A is the algebra H_{gr} for I_{2}(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, {ξ_{1}^{k}ξ_{2}^{−k}(1, 2) | 0 ≤ k ≤ r − 1}, and so k_{s} = k_{ℓ}.

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 a_{g} can be nonzero is {t_{ξ_{1}^{k}ξ_{2}^{r/2−k}(1,2)} | 0 ≤ k < r}. Thus, in the algebra A,

[v_{1}, v_{2}] = β ∑_{k}(t_{ξ_{1}^{2k}ξ_{2}^{r/2−2k}(1,2)} − t_{ξ_{1}^{r/2−2k}ξ_{2}^{2k}(1,2)}), where β = a_{ξ_{2}^{r/2}(1,2)}(v_{1}, v_{2}).

5. A different graded Hecke algebra for G(r, 1, n)

The classification of graded Hecke algebras for complex reflection groups in Section 2 shows that there do not exist graded Hecke algebras A ≅ S(V) ⊗ ℂG for the groups G = G(r, 1, n), r > 2, n > 3. In this section, we define a different “semidirect product” of the symmetric algebra S(V) and the group algebra ℂG for the groups G(r, 1, n). These algebras are not graded Hecke algebras in the sense of Section 1, but they do have a structure similar to what we would expect from experience with graded Hecke algebras for real reflection groups. Is it possible that there is a general definition of graded Hecke algebras, different from that given in Section 1, which includes the algebras defined below as examples for the groups G(r, 1, n)?

We shall use the notation for the groups G(r, 1, n) as in Section 2B so that the group G(r, 1, n) is acting by monomial matrices on a vector space V of dimension n with orthonormal basis {v_{1}, …, v_{n}}. Let s_{i} denote the permutation (i, i + 1) ∈ G(r, 1, n).

Define H_{r,1,n}^{∗} to be the algebra generated by the group algebra ℂG(r, 1, n) and V with relations

v_{i}v_{j} = v_{j}v_{i},  for all 1 ≤ i, j ≤ n,
t_{ξ_{i}}v_{j} = v_{j}t_{ξ_{i}},  for all 1 ≤ i, j ≤ n,
t_{s_{i}}v_{k} = v_{k}t_{s_{i}},  if k ∉ {i, i + 1},

t_{s_{i}}v_{i+1} = v_{i}t_{s_{i}} + ∑_{ℓ=0}^{r−1} t_{ξ_{i}^{ℓ}ξ_{i+1}^{−ℓ}} ,  for 1 ≤ i ≤ n − 1.

(5.1)

The following proposition establishes an “evaluation homomorphism” for the algebras H_{r,1,n}^{∗} which is a generalization of the homomorphism in (4.5).

Proposition 5.2. Define elements v̄_{k} in the group algebra ℂG(r, 1, n) by setting v̄_{1} = 0 and

v̄_{k} = \frac{1}{r} ∑_{i<k} ∑_{0≤ℓ≤r−1} t_{ξ_{i}^{ℓ}ξ_{k}^{−ℓ}(i,k)}, for 2 ≤ k ≤ n.

Then there is a surjective algebra homomorphism

\begin{matrix}H_{r,1,n}^{∗} & ⟶ & ℂG(r, 1, n) \\ t_{g} & ⟼ & t_{g} \\ v_{k} & ⟼ & v̄_{k} \end{matrix}

Proof. We must check that the defining relations (5.1) of H_{r,1,n}^{∗} hold with the v_{k} replaced by the v̄_{k}.

For each 1 ≤ k ≤ n, let

z̄_{k} = v̄_{1} + ⋯ + v̄_{k} = \frac{1}{r} ∑_{{1≤i<j≤k \atop 0≤ℓ≤r−1}} t_{ξ_{i}^{ℓ}ξ_{j}^{−ℓ}(i,j)}.

Then, for each k, z̄_{k} ∈ Z(ℂG(r, 1, k)) since it is the sum of the elements of the conjugacy class of reflections t_{ξ_{i}^{ℓ}ξ_{j}^{−ℓ}(i,j)} in G(r, 1, k). So z̄_{k} commutes with z̄_{1}, …, z̄_{k} and therefore z̄_{1}, …, z̄_{n} commute. Since v̄_{k} = z̄_{k} − z̄_{k−1}, it follows that v̄_{1}, …, v̄_{n} also commute.

If m > k then t_{ξ_{m}} clearly commutes with z̄_{k}. If m ≤ k then t_{ξ_{m}} commutes with z̄_{k} since z̄_{k} ∈ Z(G(r, 1, k)). So t_{ξ_{m}} commutes with z̄_{1}, …, z̄_{n} and hence with v̄_{1}, …, v̄_{n}.

Since

t_{s_{k}} v̄_{k}t_{s_{k}} = t_{s_{k}} ( ∑_{{i<k \atop 0≤ℓ≤r−1}} t_{ξ_{i}^{ℓ}ξ_{k}^{−ℓ}(i,k)}) t_{s_{k}} = ∑_{{i<k \atop 0≤ℓ≤r−1}} t_{ξ_{i}^{ℓ}ξ_{k+1}^{−ℓ}(i,k+1)}

= ∑_{{i<k+1 \atop 0≤ℓ≤r−1}} t_{ξ_{i}^{ℓ}ξ_{k+1}^{−ℓ}(i,k+1)} − ∑_{0≤ℓ≤r−1} t_{ξ_{k}^{ℓ}ξ_{k+1}^{−ℓ}(k,k+1)}

= v̄_{k+1} − ∑_{0≤ℓ≤r−1} t_{ξ_{k}^{ℓ}ξ_{k+1}^{−ℓ}} t_{s_{k}},

it follows that

v̄_{k}t_{s_{k}} = t_{s_{k}} v̄_{k+1} − ∑_{0≤ℓ≤r−1} t_{s_{k}}t_{ξ_{k}^{ℓ}ξ_{k+1}^{−ℓ}}t_{s_{k}} = t_{s_{k}} v̄_{k+1} − ∑_{0≤ℓ≤r−1} t_{ξ_{k}^{−ℓ}ξ_{k+1}^{ℓ}} = t_{s_{k}} v̄_{k+1} − ∑_{0≤ℓ≤r−1} t_{ξ_{k}^{ℓ}ξ_{k+1}^{−ℓ}}.

6. References

[Bou] N. Bourbaki, Groupes et algebres de Lie, Chapt. IV-VI, Masson, Paris, 1981.

[Ca] R.W. Carter, Conjugacy classes in the Weyl group, Compositio Math. 25 (1972), 1-59.

[Dr] V.G. Drinfeld, Degenerate affine Hecke algebras and Yangians, Funct. Anal. Appl. 20 (1986), 58-60.

[G+] M. Geck, G. Hiss, F. Lübeck, G. Malle, and G. Pfeiffer, CHEVIE – A system for computing and processing generic character tables for finite groups of Lie type, Weyl groups and Hecke algebras, AAECC, 7 (1996), 175-210.

[Hu] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics 29, Cambridge University Press, Cambridge, 1990.

[Lu] G. Lusztig, Affine Hecke algebras and their graded version, J. Amer. Math. Soc. 2 (1989), 599-635.

[Lu2] G. Lusztig, Cuspidal local systems and graded Hecke algebras I, Inst. Hautes Études Sci. Publ. Math. 67 (1988), 145-202.

[Mac] I.G. Macdonald, Symmetric functions and Hall polynomials, Second edition, Oxford Mathematical Monographs, Oxford University Press, New York, 1995.

[OS1] P. Orlik and L. Solomon, Coxeter arrangements, Proc. Symp. Pure Math. 40 (1983), 269-291.

[OS2] P. Orlik and L. Solomon, Arrangements defined by unitary reflection groups, Math. Ann. 261 (1982), 339-357.

[OT] P. Orlik and H. Terao, Arrangements of hyperplanes, Springer-Verlag, Berlin-Heidelberg, [S+] Martin Schnert et.al., GAP – Groups, Algorithms, and Programming, Lehrstuhl D für

Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, fifth edition, 1995.

[ST] G.C. Shephard and J.A. Todd, Finite Unitary Reflection Groups, Canad. J. Math. 6 (1954), 274-304.

[St1] R. Steinberg, Differential equations invariant under finite reflection groups, Trans. Amer. Math. Soc. 112 (1964), 392-400.

[St2] R. Steinberg, Endomorphisms of linear algebraic groups, Mem. Amer. Math. Soc. 80 (1968), 1-108.