Last update: 22 January 2014
The classification of graded Hecke algebras for complex reflection groups in Section 2 shows that there do not exist graded Hecke algebras for the groups In this section, we define a different “semidirect product” of the symmetric algebra and the group algebra for the groups 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
We shall use the notation for the groups as in Section 2B so that the group is acting by monomial matrices on a vector space of dimension with orthonormal basis Let denote the permutation
Define to be the algebra generated by the group algebra and with relations The following proposition establishes a "evaluation homomorphism" for the algebras which is a generalization of the homomorphism in (4.5).
Define elements in the group algebra by setting and Then there is a surjective algebra homomorphism
We must check that the defining relations (5.1) of hold with the replaced by the
For each let Then, for each since it is the sum of the elements of the conjugacy class of reflections in So commutes with and therefore commute. Since it follows that also commute.
If then clearly commutes with If then commutes with since So commutes with and hence with
Since it follows that
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.