Orbits of cuspidal types on ${\text{GL}}_p({\mathcal{O}}_F)$

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.

26 pages, improved exposition