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Unipotent representations and cohomological induction. (English) Zbl 0822.22009
Eastwood, Michael (ed.) et al., The Penrose transform and analytic cohomology in representation theory. AMS-IMS-SIAM summer research conference, June 27 - July 3, 1992, South Hadley, MA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 154, 47-70 (1993).
Let $$G$$ be a real semisimple Lie group. This paper discusses the “cohomological induction” construction of representations of $$G$$. The author sketches the most important results about the duality of cohomologically induced modules and the existence of invariant hermitian forms on modules in the “middle degree”. Then he discusses the “positivity” conditions which imply the vanishing of all cohomologically induced modules except the one in the “middle degree” and irreducibility and unitarity of the latter. The final section reviews the author’s work on various generalizations of these results and their relationship to still mysterious “unipotent representations”. A number of illuminating examples of such representations is discussed.
For the entire collection see [Zbl 0780.00026].

##### MSC:
 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) 22E46 Semisimple Lie groups and their representations 22E70 Applications of Lie groups to the sciences; explicit representations 14F05 Sheaves, derived categories of sheaves, etc. (MSC2010) 14F17 Vanishing theorems in algebraic geometry 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 32C36 Local cohomology of analytic spaces 32Q45 Hyperbolic and Kobayashi hyperbolic manifolds