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Special K-types, tempered characters and the Beilinson-Bernstein realization. (English) Zbl 0655.22010
Let G be a connected semisimple Lie group and K its maximal compact subgroup. In his paper [Ann. Math., II. Ser. 109, 1-60 (1979; Zbl 0424.22010)] D. Vogan introduced the notion of the lowest K-type of an irreducible admissible representation of G. He also gave a classification of irreducible representations in terms of these K-types. Using theory of D-modules, A. Beilinson and J. Bernstein gave another classification of these representations in terms of the orbits Q of the complexification of K in the flag variety X of the complexified Lie algebra $${\mathfrak g}$$ of G, and K-equivariant connections $$\tau$$ on Q [C. R. Acad. Sci., Paris, Sér. I 292, 15-18 (1981; Zbl 0476.14019)]. In this classification, every irreducible representation is isomorphic to global sections of an irreducible D-module $$L_{Q,\tau}$$, which is the unique irreducible submodule of a “standard” module $$I_{Q,\tau}.$$
The author defines a “special” K-type of a “standard” module $$\Gamma (X,I_{Q,\tau})$$ in geometric terms, and proves that these K-types lie in the unique irreducible submodule $$\Gamma (X,L_{Q,\tau})$$. Finally, he proves that they are equal to Vogan’s lowest K-types.
Reviewer: D.Miličić

##### MSC:
 22E46 Semisimple Lie groups and their representations 22E47 Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) 14M17 Homogeneous spaces and generalizations 14L17 Affine algebraic groups, hyperalgebra constructions
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