The Stone-von Neumann-Mackey Theorem for locally compact abelian groups is proved using the Peter-Weyl theorem and the theory of Fourier transforms for finite dimensional real vector spaces. A theorem of Pontryagin and van Kampen on the structure of locally compact abelian groups (which is evident in any particular case) is assumed.

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).

This article gives a fairly self-contained treatment of the basic facts about the Iwahori-Hecke algebra of a split p-adic group, including Bernstein's presentation, Macdonald's formula, the Casselman-Shalika formula, and the Lusztig-Kato formula.

We develop a simple algebraic approach to the study of the Weil representation associated to a finite abelian group. As a result, we obtain a simple proof of a generalisation of a well-known formula for the absolute value of its character. We also obtain a new result about its decomposition into irreducible representations. As an example, the decomposition of the Weil representation of Sp_2g(Z/NZ) is described for odd N.

A characterization of the maximal abelian sub-algebras of matrix algebras that are normalized by the canonical representation of a finite Heisenberg group is given. Examples are constructed using a classification result for finite Heisenberg groups.

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 cuspidality, which arises in the construction of supercuspidal representations of GL_n(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 GL_n(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 GL_n(O_k) is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for GL₄(O₂) are constructed. Not all these representations are strongly cuspidal.

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.

Inductive algebras for the irreducible unitary representations of the universal cover of the group of unimodular two by two matrices are classified. The classification of homogeneous shift operators is obtained as a direct consequence. This gives a new approach to the results of Bagchi and Misra.

We announce ultrametric analogues of the results of Kleinbock-Margulis for shrinking target properties of semisimple group actions on symmetric spaces. The main applications are S-arithmetic Diophantine approximation results and logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on trees.

Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are studied in the spirit of the treatment of the spherical harmonics by Stein and Weiss. As a special case, a simple self-contained proof of Laplace's integral for a Legendre function is obtained.

An element x of a finite group G is said to be p-regular if its order is not divisible by p. Brauer gave several proofs of the fact that the number of isomorphism classes of irreducible representations of G over an algebraically closed field of characteristic p is the same as the number of conjugacy classes in G that consist of p-regular elements. One such proof is presented here.

Let A be a local commutative principal ideal ring. We study the double coset space of GL(n,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 Aⁿ. If k is the length of the ring, we determine for which of the pairs (n,k) the double coset space depend 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.

This article gives conceptual statements and proofs relating parabolic induction and Jacquet functors on split reductive groups over a non-Archimedean local field to the associated Iwahori-Hecke algebra as tensoring from and restricting to parabolic subalgebras. The main tool is Bernstein's presentation of the Iwahori-Hecke algebra.

Let G be a split semisimple group over a finite field F_q, F the field F_q(t) of rational functions in t with coefficients in F_q and A the adèles of F. We describe the irreducible automorphic representations of G(A) which have non-zero vectors invariant under Iwahori subgroups at two places and under maximal compact subgroups at all other places in terms of the irreducible square-integrable representations of an Iwahori-Hecke algebra associated to G and the Satake isomorphism.

Let G be a split reductive group over a finite field F_q. Let F=F_q(t) and let A denote the adèles of F. We show that every double coset in G(F)\G(A)/K has a representative in a maximal split torus of G. Here K is the set of integral adelic points of G. When G ranges over general linear groups this is equivalent to the assertion that any algebraic vector bundle over the projective line is isomorphic to a direct sum of line bundles.

Let G be a split adjoint group defined over F_q, let F_q(t), and let A be the adèles of F. We describe the local constituents at two points of automorphic representations of G in the discrete part of L²(G(F)\G(A)) which have vectors invariant under Iwahori subgroups at these two points and are unramified at all other points.

Let G be a split semisimple group over a finite field F_q, let F = F_q (t), and let A denote the adèles of F. For all the irreducible representations of G(A) occurring in the discrete part of L²(G(F)\G(A)) which have vectors invariant under Iwahori subgroups at two places of F and maximal compact subgroups at all other places, we describe the local constituents at those two places in terms of the irreducible square integrable representations of an Iwahori Hecke algebra. We include proofs of certain well known results about the classification of principal G-bundles on the projective line which we use in our calculations.