Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Correspondence between Lie algebra invariant subspaces and Lie group invariant subspaces of representations of Lie groups
HTML articles powered by AMS MathViewer

by Joel Zeitlin PDF
Trans. Amer. Math. Soc. 167 (1972), 227-242 Request permission

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 $\mathfrak {g} \leqq 2$.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 22E60, 47A15
  • Retrieve articles in all journals with MSC: 22E60, 47A15
Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 167 (1972), 227-242
  • MSC: Primary 22E60; Secondary 47A15
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0297928-X
  • MathSciNet review: 0297928