The regular representations of ${\mathrm{GL}}_N$ over finite local principal ideal rings

Abstract

Let $\mathfrak{o}$ be the ring of integers in a non-Archimedean local field with finite residue field, $\mathfrak{p}$ its maximal ideal, and $r\ge 2$ an integer. An irreducible representation of the finite group $G_r={\mathrm{GL}}_N(\mathfrak{o}/{\mathfrak{p}}^r)$ is called regular if its restriction to the principal congruence kernel $K^{r-1}=1+{\mathfrak{p}}^{r-1}{\mathrm{M}}_N(\mathfrak{o}/{\mathfrak{p}}^r)$ consists of representations whose stabilisers modulo $K^1$ are centralisers of regular elements in ${\mathrm{M}}_N(\mathfrak{o}/\mathfrak{p})$. The regular representations form the largest class of representations of $G_r$ which is currently amenable to explicit construction. Their study, motivated by constructions of supercuspidal representations, goes back to Shintani, but the general case remained open for a long time. In this paper we give an explicit construction of all the regular representations of $G_r$.

20 pages

References in corpus (5)

• The algebraisation of higher Deligne--Lusztig representations
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• Unramified representations of reductive groups over finite rings
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Cited by in corpus (3)

• A multiplicity one theorem for groups of type $A_n$ over discrete valuation rings
• Representations of ${\mathrm{GL}}_N$ over finite local principal ideal rings - an overview
• The stable representations of ${\mathrm{GL}}_N$ over finite local principal ideal rings