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.
Categories: math.RT