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Speakers, Titles and Abstracts:
Barbasch, Dan (Cornell)
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Dirac cohomology and unipotent representations:
Dirac cohomology is an invariant of admissible representations
generalizing (g,K) cohomology, first introduced and studied by Huang
and
Pandzic. In this talk I will present results joint with P. Pandzic on
computing this invariant for unitary representations for complex and
real groups.
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Frenkel, Igor (Yale)
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Vertex operator algebras and semi-infinite cohomology
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Gaitsgory, Dennis (Harvard)
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A certain chiral algebra appearing in the geometric Langlands
correspondence
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Garland, Howard (Yale)
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A Siegel-Weil Theorem for Loop Groups:
We will discuss automorphic forms on arithmetic quotients such as
G/GZ where G is the group of real points of a loop group
and GZ the group of Z rational points. In
this setting there is a theory of fundamental domains from Siegel
sets, Eisenstein series, theta series and a Siegel-Weil Theorem,
extending the classical results of of Siegel and Weil. This is joint
work with Yongchang Zhu.
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Howe, Roger (Yale)
Kobayashi, Toshiyuki (Tokyo)
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Branching problems for Zuckerman's derived functor modules
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Kostant, Bert (MIT)
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On three exotic finite subgroups of E8 and polynomial invariants of degrees 30,24, and 20:
(joint work with N.Wallach). Let g be a complex simple Lie algebra
and let G be the adjoint group of g. Let h be the Coxeter number of
g. Some time ago I conjectured that if q= 2h+1 is a prime power then
then the finite simple group L2(q) embeds into G. With the help of
computers, in a number of the cases, this has been shown to be
true. The most sophisticated case is when G = E8. Here q=61. This
embedding was first computer established by Cohen-Griess and later
without computer by Serre. Griess-Ryba also later (computer) proved
that L2(49) and L2(41) embed into E8.
Write the three power primes 61, 49, 41 as qk where k= 30, 24, 20 so
that qk = 2, k +1. In a 1959 paper I related, for any simple g, the
Coxeter element with the principal nilpotent element in g. Tony
Springer, in a 1974 paper, extending my result in the special case of
E8, established a similar connection, between three nilpotent
elements, ek in g, and three (regular) elements of the Weyl group
σk. The order of σk is k. Using some beautiful properties
of σk the main result in this talk is the establishent of a
clear cut connection between Springer's result, on one hand, with the
Griess-Cohen-Ryba embedding L2(qk) in E8 on the other.
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Lisi, Garrett (FQXi)
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E8 Theory:
All of the gravitational and standard model particle fields of physics
may be described as parts of a superconnection valued in the Lie
algebra of a noncompact real form of E8, with dynamics described by
its curvature. This algebra of standard model fields and its embedding
in E8 may be exhibited explicitly by a matrix representation, and
schematically using weight diagrams. Several open questions with this
model remain, including the mathematical interpretation of the
superconnection -- which has an analogue in the BRST formulation of
gauge theories.
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Lusztig, George (MIT)
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Cyclic quivers and antiorbital complexes
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Penkov, Ivan (Jacobs University,
Bremen)
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Locally semisimple and maximal subalgebras of sl(∞), o(∞), sp(∞):
I will discuss an infinite-dimensional generalization of the classical
work of Malcev and Dynkin on semisimple and maximal subalgebras of
semisimple Lie algebras. In particular, I will show that any locally
semisimple subalgebra of sl(∞), o(∞), sp(∞) is in
fact semisimple, and will describe all simple and all maximal
subalgebras of sl(∞), o(∞), sp(∞). This description
should play a role in a future theory of (g,K)-modules for
g= sl(∞), o(∞), sp(∞).
This is joint work with I. Dimitrov and is dedicated to Gregg
Zuckerman's 60th birthday.
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Sally, Paul (Chicago)
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Supertempered Distributions on Reductive Groups:
We discuss the Fourier transform of orbital integrals and their
relation to the supertempered distributions of Harish-Chandra. This
work is closely related to the work of Zuckerman. At the end, we
outline the current status of this problem for p-adic groups.
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Schmid, Wilfried (Harvard)
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Hodge structures and unitary representations:
To understand the irreducible unitary representations of a reductive
Lie group G, it suffices to consider Harish Chandra modules whose
infinitesimal character is real, relative to the weight lattice. As
Vogan has pointed out, in this situation the Harish Chandra module
carries a hermitian bilinear form which is infinitesimally invariant
under U, a compact real form of the complexification of G. It is
related to the infinitesimally G-invariant hermitian bilinear form
when that exists, especially transparently in the equal rank
case. Vogan has used this relationship to formulate a conjecture on
the signature character. I shall describe a conjecture to the effect
that Saito's theory of mixed Hodge modules can be used to realize the
infinitesimally U-invariant hermitian form geometrically, in terms of
an infinite dimensional polarized Hodge structure on the
Harish-Chandra module. The conjecture would imply Vogan's conjecture
on the signature characters, but would also have other
consequences. This is joint work with Kari Vilonen.
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Serganova, Vera (Berkeley)
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On the category of integrable modules over direct limits of classical Lie
algebras:
Let g be a direct limit of finite-dimensional reductive Lie
algebras. The category of integrable g-modules is a natural
generalization
of the category of finite-dimensional modules. It is easy to see
however
that it is not semi-simple. We show that this category has enough
injective objects, injective envelope of a simple module can be
obtained
by taking integrable part of the dual module. Inside the category of
integrable module we construct a natural semi-simple subcategory of
modules with finite weight multiplicities. The latter category is
unfortunately rather small - for instance, it does not include the
adjoint
representation. Our last result is a description of a subcategory
closed
under * and such that every module has a finite Loewy length.
This is a joint work with I. Penkov.
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Speh, Birgit (Cornell)
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Restrictions of Unitary Representations: Examples and Applications to
Automorphic Forms:
I will discuss in the first part the restriction of representations of
U(p,q) with nontrivial (g,K)-cohomology to U(r)xU(p-r,q) and
applications to the cohomology of discrete groups. In the second part
I will discuss the restriction of complementary series representations
of SO(n,1) to SO(n-1,1) and use it to show that certain
representations are isolated in the unitary dual.
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Stein, Elias (Princeton)
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Singular integrals, old and new:
Singular integrals have long played a significant role in
analysis, including in the theory of the Cauchy integral,
pseudo-differential operators, and intertwining oprators for unitary
representations. An important tool on their study has been the
Calderon-Zygmund "paradigm". After sketching some of this background,
we will describe some recent results involving a new algebra of operators
and the modification of the CZ paradigm needed to study it. This algebra
arises naturally as the resolution of the conflict between underlying
"elliptic" and "sub-elliptic" geometries.
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Trapa, Peter (Univeristy of Utah)
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Functors between representations of real and p-adic groups:
For general semisimple Lie groups, Zuckerman was the first to express
the trivial character as an alternating sum of characters of
parabolically induced representations. (Actually he obtained far more
general character identities of this sort.) Similar formulas hold for
representations of reductive p-adic groups. In this talk, we
introduce functors between representations of classical real and
p-adic groups which, among other things, relate these various
character identities. This is joint work with Dan Ciubotaru.
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Vogan, David (MIT)
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The translation principle and Hermitian forms:
Early in the 1970s, a number of mathematicians understood that the
existence of finite-dimensional representations of a real semisimple
group G imposed a lot of structure on the infinite-dimensional
representations. This idea was beautifully formalized by Jens-Carsten
Jantzen (working mostly with highest weight modules) and by Gregg
Zuckerman (working mostly with Harish-Chandra modules); it is now
known as the Jantzen-Zuckerman translation principle, since it allows
one to "translate" information about one representation to another.
I'll recall what the translation principle says, and how it allows one
to attach a finite-dimensional Weyl group representation to an
infinite-dimensional irreducible representation. Then I'll recall the
(still not perfectly understood!) process of refining that Weyl group
representation to a Hecke algebra representation, leading to the
Kazhdan-Lusztig algorithms for understanding irreducible characters.
Finally (the only new part) I'll talk about a further refinement of
the Hecke algebra representation which sheds light on signatures of
invariant Hermitian forms. This new part is joint work in progress
with Gregg's former student Jeffrey Adams, and Marc van Leeuwen, Peter
Trapa, and Wai Ling Yee.
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Willenbring, Jeb (University of
Wisconsin, Milwaukee)
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Tensor powers of the oscillator representation and the spherical principal series of GL(n,R):
Let n1, n2, ... , nr be positive integers, and set n = n1 + ... +
nr. A difficult problem in finite dimensional representation theory
is to provide a combinatorial description of the branching rule from K
= O(n) (the orthogonal group) to the block diagonally embedded
subgroup M = O(n1) x ... x O(nr). Among the known results is an
expression for the branching multiplicities involving the
Littlewood-Richardson coefficients. However, these formulas are valid
only within a certain "stable range" requiring that n1, ..., nr are
large with respect to the data describing the K-representation.
We review how this multiplicity formula can be deduced, via dual
pairs, from the decomposition of a tensor power of an irreducible,
infinite dimensional, representation of the Lie algebra sp2m(R).
We emphasize the special case of finding the dimension of the
M-invariant subspace of an irreducible K-representation. Although
outside the stable range, one may consider the case when n1 = ... =
nr = 1, which corresponds to the problem of decomposing a tensor
power of the oscillator representation. This situation is of
particular interest as it sheds light on the multiplicity of K-types
in the spherical principle series of GL(n, R).
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