Abstract. 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.

We define a new notion of cuspidality for representations of GLn 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 cuspidality, which arises in the construction of supercuspidal representations of GLn(F ). We show that strongly cuspidal representations share many features of cuspidal representations of finite general linear groups. In the function field case, we show that the construction of the representations of GLn(o k ) for k ≥ 2 for all n is equivalent to the construction of the representations of all the groups G λ . A functional equation for zeta functions for representations of GLn(o k ) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL 4 (o 2 ) are constructed. Not all these representations are strongly cuspidal.

A notion of degeneration of elements in groups is introduced. It is used to parametrize the orbits in a finite abelian group under its full automorphism group by a finite distributive lattice. A pictorial description of this lattice leads to an intuitive self-contained exposition of some of the basic facts concerning these orbits, including their enumeration. Given a partition $\lambda$, the lattice parametrizing orbits in a finite abelian p-group of type $\lambda$ is found to be independent of p. The order of the orbit corresponding to each parameter, which turns out to be a polynomial in p, is calculated. The description of orbits is extended to subquotients by certain characteristic subgroups. Each such characteristic subquotient is shown to have a unique maximal orbit.Comment: 14 pages, 5 figure

We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).Comment: 23 page

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 determine for which of the pairs (n, k) the double coset space depends on the ring in question. For n = 3, we give a complete parametrisation of the double coset space and provide estimates on the rate of growth of the number of double cosets.1.1. Related problems. Let P 1 , P 2 < G be finite groups. Let ρ i = Ind G P i 1 = C[G/P i ] be the representation of G induced from the trivial representation of P i over C. The module of intertwining operators Hom G (ρ 1 , ρ 2 ) can be identified with the subalgebra C[P 2 \G/P 1 ] of left-(P 2 , P 1 )-invariant elements in the group algebra C[G] via the map C[P 2 \G/P 1 ] → Hom(ρ 1 , ρ 2 ) given by f → T f for each f ∈ C[P 2 \G/P 1 ], 2000 Mathematics Subject Classification. 15A33,15A21.

Abstract. We give a closed formula for the number of partitions λ of n such that the corresponding irreducible representation V λ of Sn has non-trivial determinant. We determine how many of these partitions are self-conjugate and how many are hooks. This is achieved by characterizing the 2-core towers of such partitions. We also obtain a formula for the number of partitions of n such that the associated permutation representation of Sn has non-trivial determinant.

The Weil representation of the symplectic group associated to a finite abelian group of odd order is shown to have a multiplicity-free decomposition. When the abelian group is p-primary, the irreducible representations occurring in the Weil representation are parametrized by a partially ordered set which is independent of p. As p varies, the dimension of the irreducible representation corresponding to each parameter is shown to be a polynomial in p which is calculated explicitly. The commuting algebra of the Weil representation has a basis indexed by another partially ordered set which is independent of p. The expansions of the projection operators onto the irreducible invariant subspaces in terms of this basis are calculated. The coefficients are again polynomials in p. These results remain valid in the more general setting of finitely generated torsion modules over a Dedekind domain.Comment: 26 pages, 3 figures Revised version, to appear in Pacific Journal of Mathematic