Regular characters of groups of type \(\mathsf{A}_n\) over discrete valuation rings. (English) Zbl 1427.20012
Summary: Let \(\mathfrak{o}\) be a complete discrete valuation ring with finite residue field \(k\) of odd characteristic. Let \(\mathbf{G}\) be a general or special linear group or a unitary group defined over \(\mathfrak{o}\) and let \(\mathfrak{g}\) denote its Lie algebra. For every positive integer \(\ell\), let \(\mathsf{K}^\ell\) be the \(\ell\)-th principal congruence subgroup of \(\mathbf{G}(\mathfrak{o})\). A continuous irreducible representation of \(\mathbf{G}(\mathfrak{o})\) is called regular of level \(\ell\) if it is trivial on \(\mathsf{K}^{\ell+1}\) and its restriction to \(\mathsf{K}^\ell/\mathsf{K}^{\ell+1} \simeq \mathfrak{g}(k)\) consists of characters with \(\mathbf{G}(\overline{k})\)-stabiliser of minimal dimension. In this paper we construct the regular characters of \(\mathbf{G}(\mathfrak{o})\), compute their degrees and show that the latter satisfy Ennola duality. We give explicit uniform formulae for the regular part of the representation zeta functions of these groups.
MSC:
| 20C15 | Ordinary representations and characters |
| 20G05 | Representation theory for linear algebraic groups |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
| 15B57 | Hermitian, skew-Hermitian, and related matrices |
| 20F69 | Asymptotic properties of groups |
| 20G25 | Linear algebraic groups over local fields and their integers |
| 20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |
| 22E50 | Representations of Lie and linear algebraic groups over local fields |
Keywords:
representations of compact \(p\)-adic groups; representation zeta functions; Ennola dualityReferences:
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