Finite Rank Homogeneous Holomorphic Bundles in Flag Spaces

Abstract

For more than forty years the study of homogeneous holomorphic vector bundles has resulted in an important source of irreducible unitary representations for a real reductive Lie group. In the mid 1950s, Harish-Chandra realized a family of irreducible unitary representations for some semisimple groups, using the global sections of homogeneous bundles defined over Hermitian symmetric spaces [6]. At about the same time Borel and Weil constructed the irreducible representations for a connected compact Lie group as global sections of line bundles defined over complex projective homogeneous spaces [3]. More than ten years later, W. Schmid in his thesis solved a conjecture by Langlands and generalized the Borel-Weil-Bott theorem to realize discrete series representations for noncompact semisimple groups [16]. This extension is nontrivial for one thing because it requires an understanding of the representations obtained on some infinite-dimensional sheaf cohomology groups.