Regular characters of classical groups over complete discrete valuation rings

Abstract

Let $\mathfrak{o}$ be a complete discrete valuation ring with finide residue field $\mathsf{k}$ of odd characteristic, and let $\mathbf{G}$ be a symplectic or special orthogonal group scheme over $\mathfrak{o}$. For any $\ell\in\mathbb{N}$ let $G^\ell$ denote the $\ell$-th principal congruence subgroup of $\mathbf{G}(\mathfrak{o})$. An irreducible character of the group $\mathbf{G}(\mathfrak{o})$ is said to be regular if it is trivial on a subgroup $G^{\ell+1}$ for some $\ell$, and if its restriction to $G^\ell/G^{\ell+1}\simeq \mathrm{Lie}(\mathbf{G})(\mathsf{k})$ consists of characters of minimal $\mathbf{G}(\mathsf{k}^{\rm alg})$ stabilizer dimension. In the present paper we consider the regular characters of such classical groups over $\mathfrak{o}$, and construct and enumerate all regular characters of $\mathbf{G}(\mathfrak{o})$, when the characteristic of $\mathsf{k}$ is greater than two. As a result, we compute the regular part of their representation zeta function.