Intertwining semisimple characters for p-adic classical groups

TL;DR

This paper establishes conditions under which intertwining semisimple characters imply conjugacy in p-adic classical groups, advancing the classification of their irreducible representations and proving a Skolem-Noether type result for Lie algebra elements.

Contribution

It introduces a geometric combinatoric condition for intertwining implying conjugacy and proves a Skolem-Noether theorem for semisimple Lie algebra elements in p-adic classical groups.

Findings

Intertwining implies conjugacy under specific geometric conditions.

Two semisimple Lie algebra elements with the same characteristic polynomial are conjugate if intertwined.

Results are key steps towards classifying cuspidal representations of p-adic classical groups.

Abstract

Let~$G$ be a unitary group of an~$\epsilon$-hermitian form~$h$ given over a nonarchimedean local field~$F_0$ of odd residue characteristic. We introduce a geometric combinatoric condition under which we prove "Intertwining implies Conjugacy" for semisimple characters of~$G$ and the general linear group of the ambient vector space of~$G$. Further we prove a Skolem-Noether result for the action of~$G$ on its Lie algebra, more precisely two Lie algebra elements of~$G$ which have the same characteristic polynomial over~$F$ must be conjugate under an element of~$G$ if there are corresponding semisimple characters which intertwine over an element of~$G$ } Let~$G$ be a unitary group over a nonarchimedean local field of odd residual characteristic. This paper concerns the study of the "wild part" of the irreducible smooth representations of~$G$, encoded in a so-called "semisimple character". We prove two fundamental results concerning them, which are crucial steps towards a classification of the cuspidal representations of~$G$. First we introduce a geometric combinatoric condition under which we prove an "intertwining implies conjugacy" theorem for semisimple characters, both in~$G$ and in the ambient general linear group. Second, we prove a Skolem--Noether theorem for the action of~$G$ on its Lie algebra; more precisely, two semisimple elements of the Lie algebra of~$G$ which have the same characteristic polynomial must be conjugate under an element of~$G$ if there are corresponding semisimple strata which are intertwined by an element of~$G$.