We compute the number of orbits of pairs in a finitely generated torsion module (more generally, a module of bounded order) over a discrete valuation ring. The answer is found to be a polynomial in the cardinality of the residue field whose coefficients are integers which depend only on the elementary divisors of the module, and not on the ring in question. The coefficients of these polynomials are conjectured to be non-negative integers.
We describe bases of the Schur algebra defined by Schur and Mendez, and explain the relationship between them. In terms of these bases, we describe its structure constants, its centre and its primitive central idempotents.
Let R be a principal ideal local ring of length two, for example, the ring $R=\mathbf Z/p^2\mathbf Z$ with $p$ prime. In this paper we develop a theory of normal forms for similarity classes in the matrix rings $M_n(R)$ by interpreting them in terms of extensions of $R[t]$-modules. Using this theory, we describe the similarity classes in $M_n(R)$ for $n\leq 4$, along with their centralizers. Among these, we characterize those classes which are similar to their transposes. Non-self-transpose classes are shown to exist for all $n>3$. When $R$ has finite residue field of order $q$, we enumerate the similarity classes and the cardinalities of their centralizers as polynomials in $q$. Surprisingly, these polynomials turn out to have non-negative integer coefficients.
Let $F$ be a non-Archimedean local field and let $E$ be a finite extension of $F$. Let $G$ be a split semisimple $F$-group. We discuss how to compare distances on the Bruhat-Tits buildings $\mathbf B_E$ and $\mathbf B_F$ of $G(E)$ and $G(F)$ respectively. We also discuss the comparison of volumes on finite volume arithmetic quotients of the buildings.
A tuple (or subgroup) in a group is said to degenerate to another if the latter is an endomorphic image of the former. In a countable reduced abelian group, it is shown that if tuples (or finite subgroups) degenerate to each other, then they lie in the same automorphism orbit. The proof is based on techniques that were developed by Kaplansky and Mackey in order to give an elegant proof of Ulm's theorem. Similar results hold for reduced countably generated torsion modules over principal ideal domains. It is shown that the depth and the description of atoms of the resulting poset of orbits of tuples depend only on the Ulm invariants of the module in question (and not on the underlying ring). A complete description of the poset of orbits of elements in terms of the Ulm invariants of the module is given. The relationship between this description of orbits and a very different-looking one obtained by Dutta and Prasad for torsion modules of bounded order is explained.
We prove positive characteristic versions of the logarithm laws of Sullivan and Kleinbock-Margulis and obtain related results in Metric Diophantine Approximation.
The centralizer algebra of a matrix consists of those matrices that commute with it. We investigate the basic representation-theoretic invariants of centralizer algebras, namely their radicals, projective indecomposable modules, injective indecomposable modules, simple modules and Cartan matrices. With the help of our Cartan matrix calculations we determine their global dimensions. Many of these algebras are of infinite global dimension.
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 of type λ, 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.
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 λ, the lattice parametrizing orbits in a finite abelian p-group of type λ 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.
In this expository article, we discuss the relation between the Gaussian binomial and multinomial coefficients and ordinary binomial and multinomial coefficients from a combinatorial viewpoint, based on expositions by Butler, Knuth and Stanley.
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.
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.
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.
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 GLn over a finite quotient Ok of the ring of integers O of a non-Archimedean local field F using geometric and infinitesimal induction functors, which involve automorphism groups Gλ 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(Ok) for k≥ 2 for all n is equivalent to the construction of the representations of all the groups Gλ. A functional equation for zeta functions for representations of GLn(Ok) 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.
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 Sp2g(Z/NZ) is described for odd N.
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 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 Fq, F the field Fq(t) of rational functions in t with coefficients in Fq 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 Fq. Let F=Fq(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 Fq, let Fq(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 Fq, let F = Fq (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.