Local Differential Geometry on the Tempered Dual of a Semisimple Lie Group

Delorme proved that the Fell topology on the tempered dual of a real semi simple group G is rather simple: roughly speaking, it is identical with the "parameter topology." The aim of this paper is to prove that the "differential geometry" of the tempered dual is very simple, too; by differential geometry, we mean three types of objects: the categories of finite length (g, K)-modules with tempered subquotients, the Extn-groups between such modules, and the deformations of such modules.