Orbits of cuspidal types on $\textrm{GL}_{p}(\mathcal{O}_{F})$

TL;DR

This paper characterizes the orbits of cuspidal types on , identifying regular orbits and illustrating that orbit data alone may not distinguish cuspidal types from other representations.

Contribution

It provides a detailed description of orbits of cuspidal types on  for prime p, including criteria for regularity and counterexamples for orbit-based classification.

Findings

Identified which orbits correspond to cuspidal types.

Determined conditions for regular orbits.

Showed orbit data alone cannot always identify cuspidal types.

Abstract

Let $F$ be a non-Archimedean local field and let $\mathcal{O}_{F}$ be its ring of integers. The orbit of an irreducible representation $\rho$ of $\mathrm{GL}_n(\mathcal{O}_F)$ is a conjugacy class in $\mathfrak{gl}_n(\mathcal{O}_F)$ attached to $\rho$ by means of Clifford's theory. We give a description of orbits of cuspidal types on $\mathrm{GL}_{p}( \mathcal{O}_{F})$, with $p$ prime. We determine which of them are regular and we provide an example which shows that the orbit of a representation does not always determine whether it is a cuspidal type or not.