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

TL;DR

This work analyzes the irreducible representations of automorphism groups of $\mathfrak{o}$-modules of type $(\ell,1^n)$ by proving that their dimensions are polynomial in the residue field size $q$ via induction on $n$. Employing Clifford theory, abelian normal subgroups, and a Heisenberg-lift strategy, the authors reduce the problem to centralisers $C_{\mathrm{GL}_m(\mathfrak{o}_1)}(\beta)$ and recursively defined subgroups $P_n$ and $T_n$, yielding explicit recurrences for their representation zeta polynomials. They derive a closed-form expression for $\mathcal{R}_{G_{(\ell,1^n)}}(\mathcal{D})$ in terms of $\mathcal{R}_{\mathrm{GL}_n(\mathfrak{o}_1)}$, $\mathcal{R}_{(\mathfrak{o}_1^n\times\mathfrak{o}_1^n)\rtimes G_{(1^n)}}$, and related subpolynomials, thereby confirming Onn's conjecture for the partition $(\ell,1^n)$ and extending known cases. The results illuminate representation growth for automorphism groups of $p$-adic module lattices and provide a framework for computing zeta polynomials in this family, with some open instances (e.g., $(2,2,1)$) remaining.

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 $q=|\mathfrak{o}_1|$.

Figures (1)

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Theorems & Definitions (19)

• Conjecture 1
• Conjecture 2
• Proposition 3
• Proposition 4
• Theorem 5
• Definition 6
• Proposition 7
• Remark 8
• Proposition 9
• Proposition 10