Poincaré series of holomorphic representations

Abstract

If V is a holomorphic representation of a Hermitian symmetric group G, we can define the Poincaré series of V by PV(t) = ∑λ (dimℂ Vλ)tλ where Vλ are the eigenspaces under the center of K, a maximal compact subgroup of G. We discuss properties of these formal power series, give explicit rational forms for some of the unitary holomorphic representations, and compute their Gelfand-Kirillov dimensions and Bernstein degrees (in the sense defined by Vogan).