Dragan Miličić

Asymptotic behavior of matrix coefficients of admissible representations

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 869 1. Generalities on reductive groups . . . . . . . . . . . . . . 873 2. The infinitesimal Cartan decomposition . . . . . . . . . . . 876 3. The t-radial components . . . . . . . . . . . . . . . . . . 880 4. Differential equations satisfied by sphereical functions . . . . . 885 5. Asymptotic behavior of spherical functions on A_ . . . . . . . 888 6. Asymptotic behavior of spherical functions along the walls of A_ . 892 7. Leading characters and growth estimates on the group . . . . . 898 8. Admissible representations and their matrix coefficients . . . . 906 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 917

Asymptotic behaviour of matrix coefficients of the discrete series

A chlorine-donating perhalogenated hydrocarbon such as fluorocarbon 133a is reacted in the vapor phase with a hydrogen-donating halogenated hydrocarbon such as fluorocarbon 113a in the presence of a catalyst to produce a first halogenated fluorocarbon product with one less chlorine and one more hydrogen than the chlorine donating perhalogenated hydrocarbon and one less hydrogen and one more chlorine than the hydrogen donating halogenated hydrocarbon. Preferred catalysts are activated carbon and especially chromium oxides and oxyfluorides. The products such as fluorocarbon 123 are useful in aerosol, refrigerant and foaming applications, and as intermediates to other chlorofluorocarbons.

Twisted Harish–Chandra Sheaves and Whittaker Modules: The Nondegenerate Case

In this paper we develop a geometric approach to the study of the category of Whittaker modules. As an application, we reprove a well-known result of B. Kostant on the structure of the category of nondegenerate Whittaker modules.

EQUIVARIANT DERIVED CATEGORIES, ZUCKERMAN FUNCTORS AND LOCALIZATION

In this paper we revisit some now classical constructions of modern representation theory: Zuckerman’s cohomological construction and the localization theory of Bernstein and Beilinson. These constructions made an enormous impact on our understanding of representation theory during the last decades (see, for example, [19]). Our present approach and interest is slightly different than usual. We approach these constructions from the point of view of a student in homological algebra and not representation theory. Therefore, we drop certain assumptions natural from the point of view of representation theorists and stress some unifying principles.

Intertwining Functors and Irreducibility of Standard Harish—Chandra Sheaves

Let g be a complex semisimple Lie algebra and σ an involution of g. Denote by t the fixed point set of this involution. Let K be a connected algebraic group and ϕ a morphism of K into the group G = Int(g) of inner automorphisms of g such that its differential is injective and identifies the Lie algebra of K with t. Let X be the flag variety of g, i.e. the variety of all Borel subalgebras in g. Then K acts algebraically on X, and it has finitely many orbits which are locally closed smooth sub varieties. The typical situation is the following: g is the complexification of the Lie algebra of a connected real semisimple Lie group G 0 with finite center, K is the complexification of a maximal compact subgroup of G 0, and σ the corresponding Cartan involution.

Variations on a Casselman–Osborne Theme

We discuss two classical results in homological algebra of modules over an enveloping algebra – lemmas of Casselman–Osborne and Wigner. They have a common theme: they are statements about derived functors. While the statements for the functors itself are obvious, the statements for derived functors are not and the published proofs were completely different from each other. First we give simple, pedestrian arguments for both results based on the same principle. Then we give a natural generalization of these results in the setting of derived categories.