Tempered modules in exotic Deligne-Langlands correspondence

TL;DR

This paper classifies tempered modules for affine Hecke algebras of type C with arbitrary parameters within the exotic Deligne-Langlands framework, linking algebraic, geometric, and combinatorial perspectives.

Contribution

It provides a new classification of tempered modules and discrete series, including those containing the sign representation, extending previous conjectures.

Findings

Classification of tempered modules for type C affine Hecke algebra.

Geometric and combinatorial descriptions of discrete series.

Connection to Weyl group module structures and previous conjectures.

Abstract

The main purpose of this paper is to identify the tempered modules for the affine Hecke algebra of type $C_n^{(1)}$ with arbitrary, non-root of unity, unequal parameters, in the exotic Deligne-Langlands correspondence in the sense of Kato. Our classification has several applications to the Weyl group module structure of the tempered Hecke algebra modules. In particular, we provide a geometric and a combinatorial classification of discrete series which contain the sign representation of the Weyl group. This last combinatorial classification was expected from the work of Heckman-Opdam and Slooten.