On Admissibility and Temperedness of Representations of Real Reductive Groups

Let G be a real Lie group with reductive Lie algebra g. We call a (g, K)-module weakly admissible if its elements are K- and 3-finite, where 3 is the center of the enveloping algebra of [g, g]C. We prove that the finitely generated weakly admissible (g, K)-modules are exactly the submodules of the "almost principal" (g, K)-modules (i.e., the K-finite subspaces of representations induced continuously from finite dimensional continuous representations of a minimal parabolic subgroup). We call attention that K is not necessarily compact; moreover, the center of the (semi-simple) connected Lie subgroup with Lie algebra [g, g] may be infinite. We redefine admissibility by calling a weakly admissible (g, K)-module admissible if its K-structure is unitarizable (then a (g, K)-module is admissible in our sense if and only if its finitely generated parts are admissible in the old sense).