Orthogonality Relations and Harmonic Forms for Semisimple Lie Groups

Let G be a connected semisimple Lie group with finite center, and suppose G contains a compact Cartan subgroup T. Certain irreducible unitary representations of G arise as spaces of harmonic forms associated to Dolbeault cohomology of line bundles over the complex homogeneous space G/T. In this work the unitary structures of these realizations are directly related to the orthogonality relations for the matrix coefficients of these representations. Using this connection, we exhibit unitary realizations of certain limits of discrete series representations of SU(2, 1) as spaces of harmonic forms.