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.
Keywords
- Parabolic Subgroup
- Maximal Compact Subgroup
- Levi Factor
- Discrete Series Representation
- Real Reductive Group
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Bratten, T. (1998). Finite Rank Homogeneous Holomorphic Bundles in Flag Spaces. In: Tirao, J., Vogan, D.A., Wolf, J.A. (eds) Geometry and Representation Theory of Real and p-adic groups. Progress in Mathematics, vol 158. Birkhäuser Boston. https://doi.org/10.1007/978-1-4612-4162-1_2
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DOI: https://doi.org/10.1007/978-1-4612-4162-1_2
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