Generalized Green functions and graded Hecke algebras

TL;DR

This paper proposes a combinatorial conjecture to parametrize irreducible tempered representations of graded Hecke algebras with unequal labels, extending Springer correspondence to type B and C root systems.

Contribution

It introduces a new combinatorial framework generalizing Springer correspondence for graded Hecke algebras with unequal labels in types B and C.

Findings

Modules have a natural grading for W_0 action

Modules are determined by central character and top degree W_0-representation

Identifies Springer correspondent as an irreducible W_0-character

Abstract

We state a conjecture which gives a combinatorial parametrization of the irreducible tempered representations with real central character of a graded Hecke algebra with unequal labels, associated to a root sytem of type B or C. This conjecture is based on a combinatorial generalization of the Springer correspondence in the classical (equal label) case. In particular, the described modules turn out to have a natural grading for the action of W_0, and are completely determined by their central character together with the W_0-representation in the top degree. This latter is an irreducible W_0-character which we call Springer correspondent.