Unitary highest weight modules and degenerate principal series

Abstract

Let G be the group ~Sp(2n;R) or U(n; n) or O*(4n); P = MN be theSiegel parabolic subgroup of G. Let \omega be the oscillator representation associatedto the dual pair (O(k); Sp(2n;R)) \subseteq Sp(2kn;R); or (U(k);U(n; n)) \subseteq Sp(4kn;R); or (Sp(k); O*(4n)) \subseteq Sp(8kn;R). In this thesis, we apply theresults of Kashiwara and Vergne on harmonic polynomials and give the descriptionof certain N-invariant tempered distributions (under \omega), where Nis the unipotent radical of the Siegel parabolic P. We then embed unitaryhighest weight modules of G into certain degenerate principal series representationsof G induced from finite dimensional representations of P explicitly.