The Representations of Automorphism Groups of $\mathfrak{o}$-modules of type $(\ell,1^n)$

Abstract

Let $\mathfrak{o}$ be the valuation ring of a non-Archimedean local field with finite residue field. We give a procedure to find the representation zeta polynomial of $\mathrm{Aut}_\mathfrak{o}(\mathfrak{o}_\ell\oplus\mathfrak{o}_1^{\oplus n})$ by induction on $n$. In particular, we show that the dimensions of the representations are given by evaluating finitely many polynomials at $q=|\mathfrak{o}_1|$.