Cohomological properties of the canonical globalizations of Harish-Chandra modules. (Consequences of theorems of Kashiwara-Schmid, Casselman, and Schneider-Stuhler) (Q1379117)

Some generalizations and consequences of the theorems of Kashiwara-Schmid (1994), Casselman (unpublished) and Schneider-Stuhler (1995) for the connected, reductive algebraic groups (Wallach, 1988, 1992) are considered. As a basis the statement is used according to which a canonical globalization of Harish-Chandra modules (Bunke, Olbrich, to appear) can be considered as coefficient modules for cohomology groups with respect to cocompact discrete subgroups (Borel, Wallach, 1980) or nilpotent Lie algebras (Bratten, 1995). Finiteness and comparison theorems for these cohomology groups are obtained. The \(n\)-cohomology (Hecht, Schmid, 1983; Hecht, Taylor, 1993), gamma-cohomology (Bunke, Olbrich, 1995) and cohomology of \(S\)-arithmetic groups (Borel, Serre, 1976) are analyzed from this point of view.