The restriction of admissible modules to parabolic subalgebras

This paper studies algebraic versions of Casselman's subrepresentation theorem. Let g be a semisimple Lie algebra over an algebraically closed field F of characteristic zero and g = f®a©nbean Iwasawa decomposition for g. Then (g, f) is said to satisfy property (n) if x\M ¥= M for every admissible (g, f )-module M. We prove that, if (g, f ) satisfies property (n), then nN ¥= N whenever N is a (g, f )-module with dim N < card F. This is then used to show (purely algebraically) that (êl(n, F), êo(n, F)) satisfies property (n). The subrepresentation theorem for §/(«) is an easy consequence of this.