Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups

Abstract:

Let G be a Lie group with Lie algebra $\mathfrak {g}$ and $\mathfrak {B} = \mathfrak {u}(\mathfrak {g})$, the universal enveloping algebra of $\mathfrak {g}$; also let U be a representation of G on H, a Hilbert space, with dU the corresponding infinitesimal representation of $\mathfrak {g}$ and $\mathfrak {B}$. For G semisimple Harish-Chandra has proved a theorem which gives a one-one correspondence between $dU(\mathfrak {g})$ 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 correspondence 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 ${\Delta _f}$, the $\Delta$ finitely transforming vectors, for $\Delta \in \mathfrak {B}$. 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 $\mathfrak {g}{\Delta _f} \subseteq {\Delta _f}$, 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 $\leqq 2$. However, this condition fails, under fairly general conditions, for $\mathfrak {g} = {N_4}$, the 4-dimensional class 3 Lie algebra. ${N_4}$ is shown to be a subalgebra of all class 3 $\mathfrak {g}$ and the condition is seen to fail for all $\mathfrak {g}$ which project onto an algebra where the condition fails. The result is then extended to cover all $\mathfrak {g}$ of class 3 with general dimension 1. Finally, it is conjectured that $\mathfrak {g}{\Delta _f} \subseteq {\Delta _f}$ for all quasisimple representations if and only if class $\mathfrak {g} \leqq 2$.