Bernstein degree and associated cycles of Harish-Chandra modules - Hermitian symmetric case

Let G̃ be the metaplectic double cover of Sp(2n,ℝ), U(p,q) or O*(2p). we study the Bernstein degrees and the associated cycles of the irreducible unitary highest weight representations of G̃, by using the theta correspondence of dual pairs. The first part of this article is a summary of fundamental properties and known results of the Bernstein degrees and the associated cycles. Our first result is a comparison theorem between the K-module structures of the following two spaces; one is the theta lift of the trivial representation and the other is the ring of regular functions on its associated variety. Secondarily, we obtain the explicit values of the degrees of some small nilpotent K_C-orbits by means of representation theory. The main result of this article is the determination of the associated cycles of singular unitary highest weight representations, which are the theta lifts of irreducible representations of certain compact groups. In the proofs of these results, the multiplicity free property of spherical subgroups and the stability of the branching coefficients play important roles.