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 $\mathrm{mod}{\mathfrak{p}}^r$ contains $a\in \mathrm{Lie}(\mathbf{G})({\mathfrak{o}}_r)$ in terms of the reduction $\mathrm{mod}\mathfrak{p}$ 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 ${\mathrm{SL}}_3(\mathfrak{o})$.

20 pages. To appear in the Indiana University Mathematics Journal