57 research outputs found
Geometric representations of GL(n,R), cellular Hecke algebras and the embedding problem
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.Comment: 18 pages, final version, to appear in JPA
From p-adic to real Grassmannians via the quantum
Let F be a local field. The action of GL(n,F) on the Grassmann variety
Gr(m,n,F) induces a continuous representation of the maximal compact subgroup
of GL(n,F) on the space of L^2-functions on Gr(m,n,F). The irreducible
constituents of this representation are parameterized by the same underlying
set both for Archimedean and non-Archimedean fields.
This paper connects the Archimedean and non-Archimedean theories using the
quantum Grassmannian. In particular, idempotents in the Hecke algebra
associated to this representation are the image of the quantum zonal spherical
functions after taking appropriate limits. Consequently, a correspondence is
established between some irreducible representations with Archimedean and
non-Archimedean origin.Comment: 24 pages, final version, to appear in Advances in Mathematic
Representations of automorphism groups of finite O-modules of rank two
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 these groups coherently. The
results obtained depend on the ring O in a very weak manner, mainly through the
degree of the residue field. In particular, a uniform description of the
irreducible representations of GL(2,O/P^k) is obtained, where P is the maximal
ideal of O.Comment: Final version, to appear in Advances in Mathematic
Quantum dimensions and their non-Archimedean degenerations
We derive explicit dimension formulas for irreducible -spherical
-representations where is the maximal compact subgroup of the
general linear group over a local field and is a closed
subgroup of such that realizes the Grassmannian of
-dimensional -subspaces of . We explore the fact that is
a Gelfand pair whose associated zonal spherical functions identify with various
degenerations of the multivariable little -Jacobi polynomials. As a result,
we are led to consider generalized dimensions defined in terms of evaluations
and quadratic norms of multivariable little -Jacobi polynomials, which
interpolate between the various classical dimensions. The generalized
dimensions themselves are shown to have representation theoretic
interpretations as the quantum dimensions of irreducible spherical quantum
representations associated to quantum complex Grassmannians.Comment: 41 pages, final version to appear in IMR
Geometric interpretation of Murphy bases and an application
In this article we study the representations of general linear groups which
arise from their action on flag spaces. These representations can be decomposed
into irreducibles by proving that the associated Hecke algebra is cellular. We
give a geometric interpretation of a cellular basis of such Hecke algebras
which was introduced by Murphy in the case of finite fields. We apply these
results to decompose representations which arise from the space of modules over
principal ideal local rings of length two with a finite residue field.Comment: Final version, to appear in JPAA, 14 page
On the unramified principal series of GL(3) over non-archimedean local fields
Let F be a non-archimedean local field and let O be its ring of integers. We
give a complete description of the irreducible constituents of the restriction
of the unramified principal series representations of GL(3,F) to GL(3,O).Comment: 16 pages, final versio
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