Homology of graded Hecke algebras

Let H be a graded Hecke algebra with complex deformation parameters and Weyl group W. We show that the Hochschild, cyclic and periodic cyclic homologies of H are all independent of the parameters, and compute them explicitly. We use this to prove that, if the deformation parameters are real, the collection of irreducible tempered H-modules with real central character forms a Q-basis of the representation ring of W. Our method involves a new interpretation of the periodic cyclic homology of finite type algebras, in terms of the cohomology of a sheaf over the underlying complex affine variety.

💡 Research Summary

The paper investigates the homological invariants of graded Hecke algebras, focusing on Hochschild, cyclic, and periodic cyclic homology. A graded Hecke algebra H is defined over a complex vector space V equipped with a finite Weyl group W and a set of complex deformation parameters c = (c_s)_{s∈S}, where S denotes the set of reflections in W. The algebra H is generated by the group algebra ℂ