Representation Theory and Mathematical Physics
Conference in honor of Gregg Zuckerman's 60th birthday
Yale University, October 24-27, 2009

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Speakers, Titles and Abstracts:

Barbasch, Dan (Cornell)

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.
Frenkel, Igor (Yale)

Vertex operator algebras and semi-infinite cohomology
Gaitsgory, Dennis (Harvard)

A certain chiral algebra appearing in the geometric Langlands correspondence
Garland, Howard (Yale)

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.
Howe, Roger (Yale)

Maxwell, Casimir, Zuckerman: abstract (pdf file)
Kobayashi, Toshiyuki (Tokyo)

Branching problems for Zuckerman's derived functor modules
Kostant, Bert (MIT)

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.

Lisi, Garrett (FQXi)

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.
Lusztig, George (MIT)

Cyclic quivers and antiorbital complexes
Penkov, Ivan (Jacobs University, Bremen)

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.

Sally, Paul (Chicago)

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.
Schmid, Wilfried (Harvard)

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.
Serganova, Vera (Berkeley)

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.
Speh, Birgit (Cornell)

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.
Stein, Elias (Princeton)

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.
Trapa, Peter (Univeristy of Utah)

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.
Vogan, David (MIT)

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.

Willenbring, Jeb (University of Wisconsin, Milwaukee)

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).