Adjoint Orbits of Matrix Groups over Finite Quotients of Compact Discrete Valuation Rings and Representation Zeta Functions

TL;DR

This paper develops methods to classify adjoint orbits of matrix groups over finite quotients of local rings and applies these to compute the representation zeta functions of certain principal congruence subgroups.

Contribution

It introduces a classification of adjoint orbits over finite quotients of local rings and uses this to compute zeta functions of principal congruence subgroups.

Findings

Classification of adjoint orbits in Lie algebras over finite quotients

Explicit computation of representation zeta functions for SL_3

New methods for orbit analysis over local ring quotients

Abstract

This paper gives methods to describe the adjoint orbits of $\mathbf{G}(\mathfrak{o}_r)$ on $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ where $\mathfrak{o}_r=\mathfrak{o}/\mathfrak{p}^r$ ($r\in\mathbb{N}$) is a finite quotient of the localization $\mathfrak{o}$ of the ring of integers of a number field at a prime ideal $\mathfrak{p}$ and $\mathbf{G}$ is a closed $\mathbb{Z}$-subgroup scheme of $\mathrm{GL}_{n}$ for an $n\in\mathbb{N}$ and such that the Lie ring $\mathrm{Lie}(\mathbf{G})(\mathfrak{o})$ is quadratic.. The main result is a classification of the adjoint orbits in $\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_{r+1})$ whose reduction $\bmod\,\mathfrak{p}^{r}$ contains $a\in\mathrm{Lie}(\mathbf{G})(\mathfrak{o}_r)$ in terms of the reduction $\bmod\mathfrak{p}$ of the stabilizer of $a$ for the $\mathbf{G}(\mathfrak{o}_r)$-adjoint action. As an application, this result is then used to compute the representation zeta function of the principal congruence subgroups of $\mathrm{SL}_{3}(\mathfrak{o})$.