Last update: 22 January 2014
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 be an dimensional vector space over and let be a finite subgroup of The group algebra of is Let be skew symmetric bilinear forms indexed by the elements of and let be the associative algebra generated by and with the additional relations where These relations allow every element to be written in the form where is the symmetric algebra of More precisely, one must fix a section of the canonical surjection from the tensor algebra of to and take the elements to be in the image of this section.
The structure of depends on the choices of the “parameters” Our goal is to determine when the algebra will be a “semidirect product” of and This idea motivates the following definition [Dri1986].
The algebra is a graded Hecke algebra for if or, equivalently, if the expression in (1.2) is unique for each A general element is a linear combination of products of elements and where is a basis of There are two straightening operations needed to put in the form (1.2): These two straightening operations correspond to the two identities in (1.1). Note that the “correct order” of is determined by the choice of the section of the projection Let be arbitrary elements of and let Applying the straightening operations to gives and applying the straightening operations in a different order gives Setting these two expressions equal gives the relation Similarly, applying the straightening operations to gives and applying the straightening operations in a different order gives These are equal if 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 Thus we have
Let be an algebra defined as in (1.1). Then is a graded Hecke algebra if and only if the identities (1.3) and (1.4) hold in
Using (1.1), the relations (1.3) and (1.4) can be rewritten in terms of the bilinear forms as for and
Let be a nondegenerate Hermitian form on For each set
Let be a finite subgroup of and let
|(b)||Suppose If and is a skew symmetric bilinear form such that then a satisfies (1.7).|
|(d)||If and then and|
|(e)||If and then, for all where is a basis of and is the composition of restricted to with the canonical projection|
(a) Consider the map given by Then and since, if then Since it follows that
(b) Let If any then (1.7) holds trivially for the skew symmetric form So assume each and write each as where and Then Since at least one of the is a linear combination of the other two. Say with Substituting and then yields and so (1.7) holds.
(c) Let and
(d) By (c), Since there exist with and so Let and be linearly independent elements of Then (1.7) implies that any element is a linear combination of and and so Thus and
(e) Write and with and Then since is skew symmetric and
The following theorem is a slightly strengthened version of statements (given without proof) in [Dri1986].
Let be a finite subgroup of and let denote the centralizer of an element in
|(a)||If is a graded Hecke algebra for then the values of are determined by the values of via the equation|
|(b)||For there is a graded Hecke algebra with if and only if where is restricted to the space In this case, is determined by its value on a basis of|
|(c)||Let be the number of conjugacy classes of such that and for all where is restricted to the space The sets corresponding to graded Hecke algebras form a vector space of dimension|
(a) is simply a restatement of (1.6).
(b) If is a graded Hecke algebra and then by Lemma 1.8d, and So is determined by its value on a basis of Suppose Then, by Lemma 1.8e, and so Note that and since, for each
If then, up to constant multiples, there is a unique skew symmetric form on which is nondegenerate on and which has Fix such a form and then define forms by for Let be any skew symmetric form on Then this collection 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 via (1.1).
(c) From (a) and (b) it follows that the sets running over all graded Hecke algebras for form a vector space. Since each of the collections constructed by (1.10) has its support on a single conjugacy class, these collections form a basis of the vector space of sets The only condition on the form is that it satisfies (1.6), which means that it is a element of
The following consequence of Theorem 1.9 will be useful for completing the classification of graded Hecke algebras for complex reflection groups.
Assume that contains for some If is a graded Hecke algebra for then for all
If then for every and if The statement then follows from Theorem 1.9b.
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.