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

This paper gives methods to describe the adjoint orbits of G(or) on Lie(G)(or) where or = o/pr (r ϵ ℕ) is a finite quotient of the completion o of the ring of integers of a number field at a prime ideal p, and G is a closed ℤ-subgroup scheme of GLn for an n ϵ ℕ such that the Lie ring Lie(G)(o) is quadratic. The main result is a classification of the adjoint orbits in Lie(G)(or+1) whose reduction modp r contains a ϵ Lie(G)(or) in terms of the reduction modp of the stabilizer of q for the G(or)-adjoint action. As an application, this result is then used to compute the representation zeta function of the principal congruence subgroups of SL₃(o).