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Representation Theory
Org: Wentang Kuo (Waterloo)
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- GERALD CLIFF, University of Alberta, Dept. of Math. & Stat. Sci.,
Edmonton, AB T6A 0M2
Realizing the local Weil representation over a number field
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Let K be a nonarchimedian local field whose residue field has
q=pm elements. Let W be the Weil representation of the
symplectic group Sp(2n,K). We show that W, considered either as
a projective representation of Sp(2n,K) or a representation of the
metaplectic group Mp(2n,K), has a model defined over the field
Q[Öp,Ö{-p}]. We use the Schrödinger model of the
Heisenberg group, having W act on locally constant, compactly
supported complex functions S(Y) on an n-dimensional K space
Y. We replace S(Y) by E-valued functions (still locally
constant, of compact support) on Y, where E is the field obtained
from Q by adjoining all p-power roots of unity. (This is in the
case that the characteristic of K is 0.) Then we use Galois
cohomology. As an application, we show that the local theta
correspondence can also be defined for representations over a number
field of a dual reductive pair.
This is joint work with David McNeilly, also of the Unversity of Alberta.
- CLIFTON CUNNINGHAM, University of Calgary, Alberta, T2N 1N4
Depth-zero Character Sheaves
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In this talk we describe the correspondence between characters of
depth-zero supercuspidal representations of p-adic groups and
coefficient systems on the Bruhat-Tits building of perverse sheaves.
Simple objects in this category are called depth-zero
character sheaves. We associated a distribution to each depth-zero
character sheaf which can be used to recover character values.
Although the correspondence does not match irreducible representations
with depth-zero character sheaves, the distributions associated to
depth-zero character sheaves appear naturally when studying
L-packets of representations. We will also indicate how depth-zero
character sheaves may be viewed as objects in a derived category of
l-adic sheaves on a rigid analytic space; this perspective
suggests how to build sheaves for the characters of a more general
class of admissible representations.
- JULIA GORDON, University of Toronto
Characters of depth-zero representations and motives
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Most characters of representations of p-adic groups elude explicit
computation. In this talk, we will discuss an approach to this
problem that is based on motivic integration, which allows us to
attach geometric objects to all "natural" p-adic integrals. As a
result, we will be able to show that the values of characters of
depth-zero supercuspidal representations of most classical groups can
be obtained by "counting points" of some geometric objects over the
residue field.
- KYU-KWAN LEE, Department of Mathematics, University of Toronto, Toronto,
Ontario M5S 3G3
Spherical Hecke algebras of GL(n) over 2-dimensional local
fields
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The classical Satake isomorphism plays an important role in the
Langlands program. In this talk we will try to generalize the theory
to the 2-dimensional local field case. More precisely, we will
construct the spherical Hecke algebra of GL(n) over a 2-dimensional
local field, and try to define an analogue of the Satake isomorphism,
using Fesenko's R((x))-valued measure. A connection to Kac-Moody
groups will also be briefly discussed.
This is joint work with Henry Kim.
- PAUL MEZO, Carleton University, 1125 Colonel By Drive, Ottawa, ON
K1S 5B6
Automorphism-invariant representations of real reductive
groups
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There is a well-established classification of the irreducbible
representations of a real reductive group in terms of discrete-series
representations and their related data. Given an group automorphism,
it is natural to wonder whether one can classify the irreducible
representations which are invariant under the automorphism in a
similar manner. We shall provide such a classification in the case of
split groups and give an indication towards applications and future
generalizations.
- FIONA MURNAGHAN, University of Toronto
Distinguished tame supercuspidal representations
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Let q be an involution of a group G. A representation of G
is said to be q-distinguished if there exists a nonzero linear
functional on the space of the representation that is invariant under
the fixed points Gq of q in G. Suppose that G is a
connected reductive p-adic group. We discuss distinguishedness of
tame supercuspidal representations of G, relative to various
involutions. As an application, we can determine when two tame
supercuspidal representations of G are equivalent, in terms of
conditions on the associated cuspidal G-data used to construct the
representations.
- MONICA NEVINS, University of Ottawa, Ottawa, ON
Branching Rules for Principal Series of GL(3)
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Let k be a p-adic field with integer ring R. We
consider the restriction of a principal series representation of
GL(3,k) to a maximal compact subgroup K = GL(3,R).
Its decomposition into irreducible subrepresentations is perplexing
and marvelous.
This is an update on a joint work with Peter Campbell (University of
Ottawa).
- WULF ROSSMANN, University of Ottawa, Ottawa
Representations of SL(2,Z) and elliptic modular
functions
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The group SL(2,Z) admits remarkable representations on
finite-dimensional spaces constructed from classical theta functions.
It turns out that these representations can be viewed as oscillator
representations of finite quotients SL(2,Z/nZ).
Particular examples of such groups are the binary polyhedral groups
corresponding to Dynkin diagrams of type D5,E6,E7,E8 by the
McKay correspondence. The purpose of the talk is to point out a
striking analogy between these representations of SL(2,Z)
and certain representations of Weyl groups on spaces of characters of
semisimple Lie groups, the characters of the Lie group playing the
role for the Weyl group which the modular elliptic functions play for
SL(2,Z).
- LOREN SPICE, University of Michigan, 430 Church St., Ann Arbor, MI
Supercuspidal characters of p-adic SLl, l a
prime
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Character computations, and in particular supercuspidal character
computations, are an important part of p-adic harmonic analysis. In
this talk, we arrive at explicit supercuspidal character formulae for
SLl over a p-adic field by evaluating an integral formula due
to Harish-Chandra. Our computations also allow us to describe
explicitly the local Langlands parameters of many supercuspidal
representations of GLl.
- FERNANDO SZECHTMAN, University of Regina, Regina, SK
The Steinberg lattice of a finite Chevalley group and its
modular reduction
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The talk will review a paper by R. Gow and touch upon a minor
contribution by the speaker.
Let G = G(F,Fq) denote the finite Chevalley group associated to
an indecomposable root system F over a finite field Fq of
characteristic p. In 1957 R. Steinberg constructed a minimal left
ideal I of the integral group algebra ZG possessing some
remarkable properties. One of these is that I is a free Z-module whose rank is the p-part of |G|; this gives rise to
an integral matrix representation of G, which viewed as a complex
representation is irreducible. Gow studies what happens to this
matrix representation when it is reduced modulo a prime. Our
contribution occurs when F is of type Cn and the reduction
is modulo 2.
- CHIAN-JEN WANG, University of Minnesota, 127 Vincent Hall, 206 Church
St. S.E., Minneapolis, MN 55455
Distinguished representations of metaplectic groups
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An automorphic representation on a metaplectic cover of GL is
called "distinguished" if it has a unique Whittaker model.
Distinguished representations can be viewed as generalizations of
classical theta functions. Using the method of the converse theorem,
Patterson and Piatetski-Shapiro constructed cuspidal distinguished
representations on the three-fold covers of GL(3). In this talk,
we will discuss recent progress toward generalizing the work of
Patterson and Piatetski-Shapiro to the case of four-fold covers of
GL(4).
- WAI LING YEE, University of Alberta
Signatures of Invariant Hermitian Forms
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Classifying the irreducible unitary representations of a reductive Lie
group may be formulated as the algebraic problem of classifying the
irreducible Harish-Chandra modules which admit a positive definite
invariant Hermitian form. It is thus of interest to study signatures
of invariant Hermitian forms and to understand how positivity can
fail. A special case, which may be a necessary first step in finding
a general answer, is the computation of the signature of the
Shapovalov form on irreducible Verma modules M(l). Computing
the signature of the Shapovalov form on irreducible highest weight
modules L(l) may provide insight into the potentially
analogous problem of computing signatures of invariant Hermitian forms
on standard limit representations and perhaps may yield some
interesting information concerning composition series of Verma modules.
- JIU-KANG YU, Purdue University
A construction of types
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I will present a joint work with Julee Kim on a construction of types.
- KAIMING ZHAO, Wilfrid Laurier University
Weight representations of higher rank Vorasoro algebras
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In this talk, I will summarize results leading to the classification
of irreducible weight modules with finite dimensional weight spaces
over higher rank Virasoro algebra. The classification for such
modules for rank one was given by O. Mathieu with a completely
different approach in 1993.
This talk is based on a joint paper with R. Lu.
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