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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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

Author: Joel Zeitlin
Journal: Trans. Amer. Math. Soc. 167 (1972), 227-242
MSC: Primary 22E60; Secondary 47A15
MathSciNet review: 0297928
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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$.

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Keywords: Universal enveloping algebra, Laplacian, nilpotent Lie algebra, analytic vector, well-behaved vector
Article copyright: © Copyright 1972 American Mathematical Society

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