Ladder representations of GL(n,Q_p)

TL;DR

This paper explores the structure of ladder representations of GL(n, Q_p) by linking them to graded Hecke algebra modules, using functors and resolutions to unify known results and reveal new connections.

Contribution

It demonstrates that the determinantal formula for ladder representations follows from BGG resolutions and connects semisimplicity, unitarity, and ladder representations.

Findings

Determinantal formula derived from BGG resolution.

Connection established between semisimplicity and unitarity.

Ladder representations characterized via graded Hecke modules.

Abstract

In this paper, we recover certain known results about the ladder representations of GL(n, Q_p) defined and studied by Lapid, Minguez, and Tadic. We work in the equivalent setting of graded Hecke algebra modules. Using the Arakawa-Suzuki functor from category O to graded Hecke algebra modules, we show that the determinantal formula proved by Lapid-Minguez and Tadic is a direct consequence of the BGG resolution of finite dimensional simple gl(n)-modules. We make a connection between the semisimplicity of Hecke algebra modules, unitarity with respect to a certain hermitian form, and ladder representations.