Comparing Translation and Jaquet functor over general linear groups

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.