Functors for unitary representations of classical real groups and affine Hecke algebras

TL;DR

This paper introduces exact functors linking Harish-Chandra modules of real classical groups to modules over affine Hecke algebras, preserving key properties like irreducibility and unitarity, thus connecting real and p-adic representation theories.

Contribution

It constructs and studies functors that map real group representations to affine Hecke algebra modules, preserving unitarity and irreducibility, and relates real and p-adic spherical unitary duals.

Findings

Functors map irreducible spherical representations to irreducible spherical modules.

Functors preserve unitarity of representations.

Establishes a functorial inclusion of real into p-adic spherical unitary duals.

Abstract

We define exact functors from categories of Harish-Chandra modules for certain real classical groups to finite-dimensional modules over an associated graded affine Hecke algebra with parameters. We then study some of the basic properties of these functors. In particular, we show that they map irreducible spherical representations to irreducible spherical representations and, moreover, that they preserve unitarity. In the case of split classical groups, we thus obtain a functorial inclusion of the real spherical unitary dual (with real infinitesimal character) into the corresponding p-adic spherical unitary dual.