arXiv subjects
Publications and source records attributed to Amritanshu Prasad.
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 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.
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.
The Stone-von Neumann-Mackey Theorem for Heisenberg groups associated to 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.
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.
We define a new notion of cuspidality for representations of $\GL_n$ over a finite quotient $\Oh_k$ of the ring of integers $\Oh$ of a non-Archimedean local field $F$ using geometric and infinitesimal induction functors, which involve automorphism groups $G_λ$ of torsion $\Oh$\nobreakdash-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 $\GL_n(\Oh_k)$ for $k\geq 2$ 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 $\GL_n(\Oh_k)$ is established for representations which are not contained in an infinitesimally induced representation. All the cuspidal representations for $\GL_4(\Oh_2)$ are constructed. Not all these representations are strongly cuspidal.
Is every locally compact abelian group which admits a Heisenberg central extension isomorphic to the product of a locally compact abelian group and its Pontryagin dual? An affirmative answer is obtained for all the commonly occurring types of abelian groups having Heisenberg central extensions, including Lie groups and certain finite Cartesian products of local fields and adeles. Furthermore, for these types of groups, it is found that the isomorphism class of the abelian group determines the Heisenberg group up to isomorphism, thereby providing a classification of such Heisenberg groups.
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.
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.
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.
The goal of these notes is to give a self-contained account of the representation theory of $GL_2$ and $SL_2$ over a finite field, and to give some indication of how the theory works for $GL_n$ over a finite field.
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 A^n. 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 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 $G(F)\bsl G(\A)/ K$ has a representative in a maximal split torus of $G$. Here $K$ is the set of integral adèlic 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.