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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

A local Peter-Weyl theorem


Author: Leonard Gross
Journal: Trans. Amer. Math. Soc. 352 (2000), 413-427
MSC (1991): Primary 22E30; Secondary 22C05
Published electronically: February 15, 1999
MathSciNet review: 1473442
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Abstract: An $Ad\, K$ invariant inner product on the Lie algebra of a compact connected Lie group $K$ extends to a Hermitian inner product on the Lie algebra of the complexified Lie group $K_{c}$. The Laplace-Beltrami operator, $\Delta $, on $K_{c}$ induced by the Hermitian inner product determines, for each number $a>0$, a Green's function $r_{a}$ by means of the identity $(a^{2} -\Delta /4 )^{-1} = r_{a} *$. The Hilbert space of holomorphic functions on $K_{c}$ which are square integrable with respect to $r_{a} (x)dx$ is shown to be finite dimensional. It is spanned by the holomorphic extensions of the matrix elements of those irreducible representations of $K$ whose Casimir operator is appropriately related to $a$.


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Additional Information

Leonard Gross
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
Email: gross@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02183-2
PII: S 0002-9947(99)02183-2
Received by editor(s): March 3, 1997
Received by editor(s) in revised form: April 18, 1997
Published electronically: February 15, 1999
Additional Notes: This work was partially supported by NSF Grant DMS-9501238.
Article copyright: © Copyright 1999 American Mathematical Society