International Conference on
Harmonic Analysis, Group Representations,
Automorphic Forms and Invariant Theory
(9 - 11 Jan 2006)

on the occasion of Professor Roger Howe’s 60th Birthday

~ Abstracts ~

The Burger-Sarnak theorem without equidistribution of Hecke points
Michael Cowling, University of New South Wales, Australia

The Burger--Sarnak theorem controls the automorphic spectrum of a group G in terms of the automorphic spectrum of a subgroup H. The Burger--Sarnak theorem was originially proved using the strong approximation theorem, but has now been simplified by others, including Clozel and Ullmo, and Oh. However, these versions still use the "equidistribution of Hecke points". This talk is about a method for obtaining the same result, but using less powerful tools, such as measure theory and functional analysis.

« Back...

When is an automorphic L-function nonvanishing in the critical strip?
Stephen Gelbart, Weizmann Institute, Israel

As is well-known, the Riemann zeta-function does not vanish in an open neighberhood of Re(s)=1 (de la Vallee Poissin, 1899). In collaboration with Erez Lapid, we have generalized this result for all Langlands-Shahidi automorphic L-functions. Our proof does not use the method of a trigonometric identity as does de la Vallee Poissin's; instead, we use generalized Eisenstein series, following Peter Sarnak in his 2002 proof for the zeta-function.

« Back...

Equivariant K-theory of affine flag manifolds and affine Grothendieck polynomials
Masaki Kashiwara, RIMS, Japan

We consider the flag manifold of affine Lie algebras as an (infinite-dimensional, not quasi-compact) scheme. It has a countable many orbits (Schubert cells) with respect to the Borel subgroup action, parametrized by the Weyl group. We consider the equivariant K-theory with respect to the Borel subgroup action, and we generalize the classical theorem that it is (as a ring) the (exponential) polynomials ring. (joint work with M. Shimozono)

« Back...

Multiplicity-free representations and visible actions on complex manifolds
Toshiyuki Kobayashi, RIMS, Japan

In this talk, I plan to present a simple principle that produces various multiplicity-free theorems for both finite and infinite dimensional representations (possibly, with continuous spectrum).

The main idea is to find under what assumption on group actions on holomorphic vector bundles, the multiplicity-free property propagates from fibers to sections.

« Back...

The Bernstein center and orbital integrals
Allen Moy, Hong Kong University of Science and Technology, Hong Kong

Suppose G=G(F) is the group of F-rational points of a reductive group defined over a p-adic field F. The Bernstein center is the algebra of invariant distributions on G which are essentially compact. An invariant distribution is essentially compact if its convolution against a locally constant compactly supported function is again locally constant compactly supported. A fundamental problem is to explicitly construct elements of the Bernstein center. We will recall and discuss an observation of Bernstein of an interesting element of the center for SL(n), and then discuss recent joint work with Tadic on the explicit construction of elements in the center.

« Back...

Automorphic distributions and L-functions
Wilfried Schmid, Harvard University, USA

Traditionally, the holomorphic continuation and functional equation of Langlands L-functions are proved in one of two ways: the method of integral transforms or the Langlands-Shahidi method. The known results are still quite limited. After a brief survey of the existing techniques, I shall a new describe new approach, developed jointly with Steve Miller, which has led to some new results.

« Back...

Cohomological induction and irreducible representations
David Vogan, MIT, USA

Zuckerman's construction for representations of a real reductive group G works like this. Let K be a maximal compact subgroup of G, with corresponding Cartan involution theta. Let q = l + u be a theta-stable parabolic subalgebra of the complexified Lie algebra g, and L the normalizer of q in G. Zuckerman's cohomological induction begins with an irreducible (l,L\cap K)-module Z and forms the "generalized Verma module"

V = U(g)\otimes_q Z,

which is a (g,L\cap K)-module. Finally he applies to V the "(Bernstein)-Zuckerman functors" L_p, which carry (g,L\cap K)-modules to (g,K)-modules. The "cohomologically induced representations" are L_p(V).

In this setting, it is quite natural to take any irreducible subquotient J of the generalized Verma module V, and to consider L_p(J). I will show that L_p(J) is always a direct sum of irreducible representations, which can be computed explicitly using the (proved) Kazhdan-Lusztig conjectures.

This work is motivated by Wai Ling Yee's recent calculation of the signatures of invariant Hermitian forms on J (in case L is abelian). The hope is that her work will in turn determine the signatures of induced forms on L_p(J).

« Back...

Whittaker models of degenerate principal series
Toshio Oshima, University of Tokyo, Japan

We study the homomorphisms of an irreducible degenerate series representation (of a real reductive Lie group G induced from a finite dimensional representation of a parabolic subgroup) into a representation induced from a character of a maximal unipotent subgroup of G. We will determine the dimension of the homomorphisms and clearify the differential equations satisfied by K-finite functions in the image.

« Back...