Last update: 22 January 2014
In [Lus1989], 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 be a finite real reflection group acting on and let be the root system of Let be a choice of simple roots in and let be the corresponding simple reflections in Let be the reflection in the root so that, for Let denote the set of positive roots in
Let be fixed complex numbers indexed by the roots satisfying This amounts to a choice of either one or two “parameters”, depending on whether all roots in are the same length or not. As in Section 1, let with and let be the symmetric algebra of Lusztig [Lus1989] defines the “graded Hecke algebra” with parameters to be the unique algebra structure on the vector space such that
|(3.2 a)||is a subalgebra of|
|(3.2 b)||is a subalgebra of and|
|(3.2 c)||for all and simple reflections in the simple roots|
Let as in (3.1). Use the notation for The element should be viewed as an element of and With this notation, let be the algebra (as in Section 1) generated by and with relations Note that is defined by the bilinear forms
The following theorem shows that the algebra satisfies the defining conditions (3.2 a-c) of the algebra
Let be a finite real reflection group and let be the algebra defined by (3.4).
|(a)||As vector spaces, (and hence, is a graded Hecke algebra).|
|(b)||If for then for all and simple reflections in|
First note that if then Thus, for For and a simple reflection, Using this equality, we obtain The two identities (3.6) and (3.8), as in (1.3) and (1.4), show that the algebra is isomorphic to
(b) This can now be proved by direct computation. If then by equation (3.4) and equation in the proof of Theorem 3.5. If and is a simple reflection then, by (3.7),
Theorem 3.5b shows that if is the graded Hecke algebra defined by (3.4), then the elements for generate a subalgebra of isomorphic to and these elements together with the satisfy the relations of (3.2 c). Since part (a) of Theorem 3.5 shows that is isomorphic to as a vector space, it follows that satisfies the conditions (3.2 a-c), relations which uniquely define the graded Hecke algebra 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 and
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.