Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups
Transactions of the American Mathematical Society
Let G be a Lie group with Lie algebra 9 and 8 = u(g), the universal enveloping algebra of 9; also let U be a representation of G on H, a Hubert space, with dU the corresponding infinitesimal representation of 9 and S. For G semisimple Harish-Chandra has proved a theorem which gives a one-one correspondence between dU(o) invariant subspaces and U(G) invariant subspaces for certain representations U. This paper considers this theorem for more general Lie groups. A lemma is proved giving such a
... respondence without reference to some of the concepts peculiar to semisimple groups used by Harish-Chandra. In particular, the notion of compactly finitely transforming vectors is supplanted by the notion of Ar, the A finitely transforming vectors, for A s ». The lemma coupled with results of R. Goodman and others immediately yields a generalization to Lie groups with large compact subgroup. The applicability of the lemma, which rests on the condition sA/SA" is then studied for nilpotent groups. The condition is seen to hold for all quasisimple representations, that is representations possessing a central character, of nilpotent groups of class ^2. However, this condition fails, under fairly general conditions, for 9 = Ni, the 4-dimensional class 3 Lie algebra. Nt is shown to be a subalgebra of all class 3 g and the condition is seen to fail for all 9 which project onto an algebra where the condition fails. The result is then extended to cover all 9 of class 3 with general dimension 1. Finally, it is conjectured that gA/çrA/ for all quasisimple representations if and only if class s = 2.