Square Integrable Representations of Semisimple Lie Groups

Abstract

Let D be a bounded symmetric domain. Let G be the universal covering group of the identity component ${A_0}(D)$ of the group of all holomorphic diffeomorphisms of D onto itself. In this case, any G-homogeneous vector bundle $E \to D$ admits a natural structure of G-homogeneous holomorphic vector bundles. The vector bundle $E \to D$ must be holomorphically trivial, since D is a Stein manifold. We exhibit explicitly a holomorphic trivialization of $E \to D$ by defining a map $\Phi :G \to {\text {GL}}(V)$ (V being the fiber of the vector bundle) which extends the classical “universal factor of automorphy” for the action of ${A_0}(D)$ on D. Then, we study the space H of all square integrable holomorphic sections of $E \to D$. The natural action of G on H defines a unitary irreducible representation of G. The representations obtained in this way are square integrable over $G/Z$ (Z denotes the center of G) in the sense that the absolute values of their matrix coefficients are in ${L_2}(G/Z)$.