Adjoint orbits of matrix groups over finite quotients of compact discrete valuation rings and representation zeta functions

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 $\mathrm{SL}_{3}(\mathfrak{o})$.