On some non-principal locally analytic representations induced by cuspidal Lie algebra representations

TL;DR

This paper constructs and analyzes non-principal locally analytic representations of split reductive p-adic groups, focusing on those induced by cuspidal Lie algebra modules, and establishes their ind-admissibility and supercuspidality criteria.

Contribution

It introduces a new class of locally analytic representations induced by cuspidal Lie algebra modules for GL(n+1), expanding the understanding of supercuspidal representations.

Findings

Representations are ind-admissible.

They satisfy the homological vanishing criterion for supercuspidality.

First construction of such non-principal locally analytic representations.

Abstract

Let $G$ be a split reductive $p$-adic Lie group. This paper is the first in a series on the construction of locally analytic $G$-representations which do not lie in the principal series. Here we consider the case of the general linear group $G=GL_{n+1}$ and locally analytic representations which are induced by cuspidal modules of the Lie algebra. We prove that they are ind-admissible and satisfy the homological vanishing criterion in the definition of supercuspidality in the sense of Kohlhaase.