Dolbeault cohomological realization of Zuckerman modules associated with finite rank representations
In the paper under review the author proves a long time standing conjecture of Zuckerman and Vogan. This conjecture states that the representations realized in spaces of Dolbeault cohomology are infinitesimally equivalent to some algebraic modules introduced by Zuckerman. The proof of the conjecture is essentially taken from the author's thesis. The use of cohomology in representation theory has a long and distinguished history, beginning with the Bott-Borel-Weil theorem and continuing with Schmid's formulation and proof of the Kostant-Langlands conjecture. Initial efforts to generalize this construction were unsuccessful. The fundamental difficulty was to show that the cohomology spaces were indeed complete topological vector spaces. To avoid the analytic difficulties Zuckerman and Vogan introduced an algebraic analog of such a cohomology, the Zuckerman modules. It was conjectured that both constructions, the analytic and the algebraic ones, coincide. The author proves: (1) The differentials in the Dolbeault complex have closed image. (2) The cohomologies are the maximal globalization of the Zuckerman modules. (3) If the inducing module admits infinitesimal character, then the cohomologies admit infinitesimal character.
scientific article; zbMATH DE number 764034