  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\otimes X/X) is called the 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 D.Vogan.
