Friday
2:30 Dubravka Ban
4:00 Matvei Libine
Saturday
9:30 Jiu-Kang Yu
11:00 Martin Weissman
2:30 Ralf Schmidt
4:00 Juliana Tymoczko
Sunday
9:30 Brooks Roberts
11:00 Kari Vilonen
Speaker:
Martin Weissman
Title: Paley-Wiener
theorems and Local L-functions
Abstract: Following
work of Harish-Chandra and
Waldspurger, Heiermann has
recently described a Paley-Wiener
theorem for the Hecke algebra
of a p-adic reductive group.
This theorem essentially identifies
the Hecke algebra with a space
of regular sections of an algebraic
variety, satisfying additional
conditions for the Weyl group
and intertwining operators.
I will describe how to rephrase
results of Godement-Jacquet
on L-functions in terms of
a Paley-Wiener theorem for
the matrix algebra of a p-adic
field. I will finish by describing
a conjectural framework linking
L-functions and Paley-Wiener
type theorems.
Speaker: Ralf Schmidt
Title: New- and Oldforms for GSp(4):
Local Theorems
Abstract: This talk
will summarize the results of a
joint project with
Brooks Roberts on local new- and
oldforms for irreducible,
admissible representations of GSp(4,F)
with trivial central
character, where F is a p-adic
field. The main feature of the
theory is that it considers vectors
fixed under a family of
congruence subgroups called the
paramodular groups. The main
results include: Characterizations
of representations with
non-zero paramodular-invariant
vectors; computation of the
dimension of spaces of paramodular-invariant
vectors at any
level for any irreducible, admissible
representation; uniqueness
at the minimal level; a complete
description of the structure of
oldforms. Additional results for
generic representations:
A computation of the zeta integral
of the newform shows that
it represents the L-factor, and
the minimal level coincides
with the exponent in the epsilon-factor
of the representation.
Speaker: Brooks Roberts
Title:
New- and Oldforms for GSp(4):
Evidence for a Global Theory
Abstract:
This talk is about joint work
with Ralf Schmidt and follows
his talk
on the local theory of new- and
oldforms for GSp(4). The local
theory
involves the local paramodular
groups. Just as the global Hecke
congruence subgroups of GL(2)
do, the global paramodular congruence
subgroups of GSp(4) arise naturally
in number theory. For example,
quotients of the Siegel upper
half-space of degree two by the
paramodular congruence subgroups
are moduli spaces of abelian
surfaces with certain polarizations.
Thus, it is reasonable to ask if
a global theory of new- and oldforms
for GSp(4) exists. In this talk
we will discuss a conjecture which
describes a theory of Siegel
modular newforms of degree two
with respect to the paramodular
congruence subgroups of GSp(4).
We will indicate how this conjecture
follows from a corresponding conjecture
in the language of
automorphic representations. Finally,
we will discuss evidence for
the conjecture derived from our
local theory and the conjectural
detailed structure of the discrete
automorphic spectrum of GSp(4).
Speaker:
Dubravka Ban
Title:
On Arthur's R-group
Abstract:
The R-group determines the
reducibility of the induced
representation and plays an
important role in the trace
formula. Classically, the R-group
is defined for square-integrable
or tempered representations,
using Plancherel measures.
The definition of the R-group
proposed by Arthur is in terms
of A-parameters and L-groups.
For tempered representations,
the classical R-group and Arthur’s R-group
should be the same, which is proved
in some cases. Arthur’s
definition also applies to some
nontempered unitary representations.
We study which of the properties
of the classical R-groups carry
over to Arthur's setting. This
is a work in progress, joint
with Chris Jantzen.
Speaker: Kari Vilonen
Title: Representation
theory and Hodge theory.
Abstract:
Speaker: Julianna Tymoczko
Title:
Geometric representations of the
symmetric group
Abstract: In 1976, Springer discovered
the remarkable fact that the symmetric
group acts naturally on the cohomology
of certain subvarieties of the
flag variety, now called Springer
fibers. Indeed, the top-dimensional
cohomology of the Springer fibers
is an irreducible representation
of the symmetric group, and each
irreducible representation can be
recovered uniquely in this way. Springer's
original construction was algebraic
but was followed by intense activity
on the part of many people to give
more intrinsically geometric explanations
for these representations.
We begin by presenting a sketch
of Springer theory using one of
these geometric descriptions. We
will then discuss representations
of the symmetric group that can
be constructed on the cohomology
of related subvarieties of the
Hessenberg variety, called Hessenberg
varieties.
Speaker: Matvei Livine
Title: Riemann_Roch-Hirzebruch
Integral Formula for Characters
of Reductive Lie Groups
Abstract (PDF)
Speaker:
Jiu-Kang Yu
Title:
Construction of tame types.
Abstract: This is a
joint work with Julee Kim.
Bernstein has given the finest
direct product decomposition
of the category of smooth representations
of a p-adic groups. Bushnell
and Kutzko introduced the notion
of "types" and
proposed that each summand in
Bernstein decomposition is
controlled by a type. But the
construction of types, known
to be analogous to the construction
of supercuspidal representations,
was done only in a few special
cases. We will show that an
obvious generalization of the
speaker's construction of supercuspidal
representations gives a fairly
general construction of types.
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