Representations of automorphism groups of finiteo-modules of rank two

Abstract

Let o be a complete discrete valuation domain with finite residue field. In this paper we describe the irreducible representations of the groups Aut(M) for any finite o-module M of rank two. The main emphasis is on the interaction between the different groups and their representations. An induction scheme is developed in order to study the whole family of these groups coherently. The results obtained depend on the ring o in a very weak manner, mainly through the degree of the residue field. In particular, a uniform description of the irreducible representations of GL 2 (o/p ) is obtained, where p is the maximal ideal of o.

Principal series for general linear groups over finite commutative rings

“…The thrust of the present paper is to introduce methods from invariant theory and monoidal categories to study the representation theory of automorphism groups of modules. It generalises, contextualises and complements several results and notions from the representation theory of automorphism groups of finite o ℓ -modules [1,12,7]; at the same time, it fits into a more general framework in tensor categories [6,10,9,11].…”

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“…A construction of explicit tight fusion frames corresponding to the remaining ten maximal TFF sequences requires a procedure similar to Example 7.2. Stone [1930] andvon Neumann [1931] for real Heisenberg groups is known as the Stone-von Neumann theorem. Mackey [1949] extended this theorem to locally compact Heisenberg groups (see Section 1C for the case that is pertinent to this paper, and [Prasad 2011] for a more detailed and general exposition).…”

“…Weil exploited the Stone-von Neumann-Mackey theorem to construct a projective representation of a group of automorphisms of the Heisenberg group, now commonly known as the Weil representation. Along with parabolic induction and the technique of Deligne and Lusztig [1976] using l-adic cohomology, the Weil representation is one of the most important techniques for constructing representations of reductive groups over finite fields (see [Gérardin 1977;Srinivasan 1979]) or local fields (see [Gérardin 1975] andMoeglin, Vignéras andWaldspurger [Moeglin et al 1987]). Tanaka [1967a;1967b] showed how the Weil representation can be used to construct all the irreducible representations of SL 2 ‫/ޚ(‬ p k ‫)ޚ‬ for odd p by looking at Weil representations associated to the abelian groups ‫/ޚ‬ p k ‫ޚ‬ ⊕ ‫/ޚ‬ p l ‫ޚ‬ for l ≤ k. However, most of the literature on Weil representations associated to finite abelian groups has focused on vector spaces over finite fields and on constructing representations of classical groups over finite fields.…”

“…It has been shown (by Aubert, Onn, Prasad and Stasinski [Aubert et al 2010] and Singla [2010]) that this problem is intricately related to the problem of understanding the representations of automorphism groups of finitely generated torsion modules over discrete valuation rings. However, explicit constructions have been available either for a very small class of representations [Hill 1995a;1995b] or for a very small class of groups [Onn 2008;Stasinski 2009;Singla 2010].…”

“…A small part of the decomposition has been explained in the general case by Cliff, McNeilly and Szechtman [Cliff et al 2003]. Maktouf and Torasso [2014;2012] have shown that the restriction of the Weil representation of a symplectic group over a p-adic field to a maximal compact subgroup or to a maximal elliptic torus is multiplicity-free and have given an explicit description of the irreducible subrepresentations.…”