Polynomial Invariants, differential operators and Eigenspace representations
by
Jing-Song Huang
Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

The aim of this talk is to describe the eigenspace representations on semisimple symmetric spaces, which is a fundamental problem in harmonic analysis on homogeneous spaces. This branch of mathematics has its root in the classical theory of Fourier series and Fourier integrals. It is closely connected with group representations theory in infinite-dimensional spaces that plays an important role in many recent developments of mathematics and in the interaction of mathematics and physics. The main content of the talk includes the description of G-invariant differential operators on a semisimple symmetric space G/H in terms of polynomial algebras and the corresponding space of eigenfunctions in terms of well understood infinite-dimensional representations.

More precisely, let a_q be the split Cartan subspace of G/H. If g does not contain any simple ideal of type E, the we can choose D₁, ... , D_r in the center Z(g) of universal enveloping algebra U(g) such that D₁, ... , D_r are algebraically independent and \bold D=C[D₁, ... , D_r] is isomorphic to S(a_q)^W(a_q). Denote by \Cal E_\lambda(G/H) the space of K-finite smooth functions on G/H such that Zf=\chi_\lambda(z)f. Let P be a \sigma-minimal parabolic subgroup and \pi _{P, \xi, \nu} be the principal series for G/H. We show that \Cal E_\lambda(G/H) is an admissible (g, K)-module and for \lambda in a^*_qC generic,

\Cal E\lambda(G/H) =~ Å w in \Cal W  Å \xi  \piP, \xi, \lambda(\xi),

where \Cal W is a set of representatives of W(a_q)/W_{K \cap H}, \xi runs over discrete series of M/w(M \cap H)w^-1. For a general \lambda, we prove the following identity for distribution characters

\Theta(\Cal E\lambda(G/H))= å w in \Cal W  å \xi  \Theta(\piP, \xi, \lambda(\xi)).

Application of these results to eigenspace representations on symmetric varieties will also be discussed.

Date received: April 1, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caex-53.