(October 3) Jeff Adler,
Towards a lifting of representations of finite reductive groups:
The problem of understanding base change for p-adic groups forces one to
consider a certain generalization of base change for finite groups. I will
give an introduction to base change, and give a partial description of this
generalization. (This is joint work with Joshua Lansky.)
(October 10) Michael Kapovich,
Saturation theorems.
I will describe a path model for geodesics in Euclidean buildings which is parallel to Littelman's path model in representation theory. As an application I will sketch a proof of the saturation theorem for SL(n) and its generalization to other Lie groups.
(October 24) Jeffrey Adams,
More fun with the character of the oscillator representation
representation: computing Gauss sums.
It is well known that Gauss sums play an important role in the
oscillator representation representation. Following unpublished notes
of Roger Howe, I will show how to go the other way, and compute Gauss
sums using the character of the oscillator representation for a finite field.
(October 31) Fiona Murnaghan,
Spherical characters of distinguished supercuspidal representations.
Let H be the group of fixed points of an involution
(automorphism of order two) of a group G. A representation
of G is said to be H-distinguished if there exists
a nonzero H-invariant linear functional on the space
of the representation. Suppose that G is a reductive
p-adic group. If an irreducible smooth representation of G
is H-distinguished, and if its contragredient (smooth
dual) is also H-distinguished, then we say the representation
is class one. Spherical characters are certain H-biinvariant
distributions on G (where H acts on G by multiplication on
the left and right) that are attached to class one representations.
Spherical characters are play an important role in harmonic
analysis on the reductive p-adic symmetric space G/H.
We will describe some basic properties of
spherical characters of class one supercuspidal
representations, and indicate how these properties
can be viewed as analogues of some properties of
ordinary characters.
We will discuss results showing that some
spherical characters vanish near the identity element,
and work in progress that describes the qualitative behaviour
of spherical characters that are nonvanishing near
the identity.
(November 7) Pramod Achar,
Derived categories of coherent sheaves in representation theory.
Perverse sheaves, which form an abelian subcategory of the derived category of
constructible sheaves, have played an important role in representation theory since
their introduction, nearly 30 years ago. Derived categories of coherent sheaves, on
the other hand, are relatively recent arrivals. I will describe some of the machinery
available in this setting, including, in particular, the Deligne-Bezrukavnikov theory
of "perverse coherent sheaves," and a new construction, the category of "staggered
sheaves." I will also discuss some known and conjectural applications of these
categories to representation-theoretic questions.
(November 28) Jonathan Rosenberg,
Electromagnetic duality and the Langlands program.
One of the most intriguing papers to appear recently is the mammoth
paper of
Witten and Kapustin with essentially the same title as this talk.
This paper in turn grew out of a much earlier paper by Montonen
and Olive (1977) suggesting how Langlands dual groups should appear
in the study of duality between field theories.
I will attempt to give a rough introduction to some of the ideas
in these papers, which work in two directions: Ideas from representation
theory may prove useful in physics, but also ideas from physics
may shed some new light on the Langlands program.
(November 28 at 3 PM) Stephen DeBacker,
Generalized r-facets: a primer.
The Bruhat-Tits building associated to a reductive p-adic group G
carries a natural "polysimplicial" decomposition. This natural
decomposition carries an enormous amount of information about the
parahoric subgroups associated to points in the building. In their
groundbreaking papers of the mid 1990s, Moy and Prasad (re)introduced
what have become known as Moy-Prasad filtration subgroups of G. The
study of generalized r-facets started as an attempt to better
understand the Moy-Prasad filtrations of a fixed depth (namely, r).
When r is zero, generalized r-facets are just the usual facets
occurring in the natural polysimplicial decomposition. In this talk,
we will discuss the utility of generalized r-facets in various
settings in representation theory and structure theory. For the
experts, I will also speculate, in a very vague way, on a direction in
which I think generalized r-facets may be, for lack of a better word,
generalized.
(December 5)Vivek Dhand, Geometric Langlands duality and forms of reductive groups.
Mirkovic and Vilonen have constructed the split form of the Langlands
dual group using perverse sheaves on the affine Grassmannian of a
complex reductive group G. We discuss variants of this result in the
non-split case. The construction of quasi-split forms involves
semi-linear equivariance with respect to a finite Galois group and the
group of outer automorphisms of G. The question of inner forms is
more delicate, and involves perverse sheaves with coefficients in a
locally constant sheaf of division algebras.
(March 5)Marty Weissman Metaplectic Tori In 1968, Langlands generalized class field theory, from the multiplicative group to arbitrary tori over local and global fields. His observations at the time were crucial in forming the conjectures of the Langlands program for arbitrary reductive groups over local and global fields. Many authors have attempted to incorporate "metaplectic groups" -- certain non-algebraic central extensions of reductive groups -- into the Langlands program, typically by relating representations of metaplectic groups to representations of related reductive groups.
In this talk, I will try to follow the historical path of Langlands, but allowing metaplectic groups throughout. Specifically, I will discuss algebraic tori and their metaplectic covers. Next, I will explain how to generalize the 1968 result of Langlands to parameterize representations of metaplectic tori (over local fields). Finally, I will describe what these observations for tori suggest for the future development of a metaplectic Langlands program.
(May 7) Jim Cogdell
A Report on Functoriality
I would like to both review past results and describe new results in
the program of establishing functoriality to GL(n) via the converse
theorem. Recently, with P-S and Shahidi, we have established the
stability of
certain gamma factors for quasi-split groups G under highly ramified
twists. This is used in the proof of functoriality from G to GL(n) to
finesse the lack of a full Local Langlands Conjecture for G.
I will review the ideas in the proof of functoriality
from G to GL(n) via the converse theorem to place our stability result
in context. Then I will outline what results follow and what we expect
to follow from this stability in terms of functoriality and global and
local applications.