Comparing Translation and Jaquet functor over general linear groups

TL;DR

This paper explores a functor connecting Harish-Chandra modules of complex general linear groups with graded Hecke algebra modules, revealing relationships between translation and Jacquet functors across different mathematical contexts.

Contribution

It demonstrates how a specific functor relates translation functors on the real side to Jacquet functors on the p-adic side, extending previous constructions.

Findings

The functor preserves standard and irreducible modules.

It is compatible with parabolic induction.

Establishes a connection between translation and Jacquet functors.

Abstract

Kei Yuen Chan and Kayue Daniel Wong constructed a functor from the category of Harish-Chandra modules of $\mathrm{GL}(n, \mathbb C)$ to the category of modules over graded Hecke algebra $\mathbb H_m$ of type A. This functor has several nice properties, such as compatible with parabolic inductions, and preserving standard and irreducible objects. Based on their results, we show this functor relates translation functor on the real side and Jacquet functor on the $p$-adic side.