Hecke modules based on involutions in extended Weyl groups

Abstract

Let X be the group of weights of a maximal torus of a simply connected semisimple group over C and let W be the Weyl group. The semidirect product W((Q ⊗ X)/X) is called an extended Weyl group. There is a natural C(v)-algebra H called the extended Hecke algebra with basis indexed by the extended Weyl group which contains the usual Hecke algebra as a subalgebra. We construct an H-module with basis indexed by the involutions in the extended Weyl group. This generalizes a construction of the author and Vogan.