Operator-valued Bessel functions, holomorphic discrete series, and harmonic representations of \(U(p,q)\)
Let \(G=U(p,q)\) and let \(K\) be a maximal compact subgroup. Then the Hermitian symmetric space \(G/K\) can be realized as \(D\), a Siegel domain of type II. In this paper the author applies his theory of operator- valued Bessel functions and Schrödinger-Fock spaces to some problems in representation theory of \(G\). He gives a new and simpler characterization of the ``reduced'' Schrödinger-Fock space, and a representation theoretic proof of a Paley-Wiener theorem on \(D\). The main result is a decomposition theorem of the harmonic (or Segal- Shale-Weil) representation of \(G\) into a sum of holomorphic discrete series; square-integrability of the latter representations is proved by means of square-integrability of the ``reduced'' Bessel functions.
scientific article; zbMATH DE number 89207
scientific article