Duality between GL(n,R) and the degenerate affine Hecke algebra for gl(n)

TL;DR

This paper establishes a precise correspondence between representations of the real general linear group and the degenerate affine Hecke algebra, revealing a duality that preserves key module structures under certain conditions.

Contribution

It introduces an exact functor linking Harish-Chandra modules for GL(n,R) to finite-dimensional modules of the degenerate affine Hecke algebra for gl(k), demonstrating structural preservation.

Findings

Functor maps standard modules to standard modules or zero

Functor maps irreducible modules to irreducible modules or zero

Provides a duality framework connecting two representation categories

Abstract

We define an exact functor $F_{n,k}$ from the category of Harish-Chandra modules for $GL(n,R)$ to the category of finite-dimensional representations for the degenerate affine Hecke algebra for $gl(k)$. Under certain natural hypotheses, we prove that the functor maps standard modules to standard modules (or zero) and irreducibles to irreducibles (or zero).