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 $\mathbf {C}$ and let $W$ be the Weyl group. The semidirect product $W((\mathbf {Q}\otimes X)/X)$ is called an extended Weyl group. There is a natural $\mathbf {C}(v)$-algebra $\mathbf {H}$ 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 $\mathbf {H}$-module with basis indexed by the involutions in the extended Weyl group. This generalizes a construction of the author and Vogan.