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… (More)
We study geometric representations of GL(n, R) for a ring R. The structure of the associated Hecke algebras is analyzed and shown to be cellular. Multiplicities of the irreducible constituents of these representations are linked to the embedding problem of pairs of R-modules x ⊂ y.
We interpolate between idempotents in the Hecke algebras associated with the Grassmann representation over different local fields. Consequently, we obtain a bi-jection between some irreducible representations geometrically defined for all local fields.
We define a new notion of cuspidality for representations of GL n over a finite quotient o k of the ring of integers o of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups G λ of torsion o-modules. When n is a prime, we show that this notion of cuspidality is equivalent to strong… (More)
Let A be a local commutative principal ideal ring. We study the double coset space of GLn(A) with respect to the subgroup of upper triangular matrices. Geometrically, these cosets describe the relative position of two full flags of free primitive submodules of A n. We introduce some invariants of the double cosets. If k is the length of the ring, we… (More)
We study a family of complex representations of the group GL n (o), where o is the ring of integers of a non-archimedean local field F. These representations occur in the restriction of the Grassmann representation of GL n (F) to its maximal compact subgroup GL n (o). We compute explicitly the transition matrix between a geometric basis of the Hecke algebra… (More)
In this paper similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly.
This paper generalizes some aspects of J. A. Green's work on the cuspidal representations of general linear groups over finite fields to general linear groups over finite quotients of discrete valuation rings.