A local Peter-Weyl theorem

Author:
Leonard Gross

Journal:
Trans. Amer. Math. Soc. **352** (2000), 413-427

MSC (1991):
Primary 22E30; Secondary 22C05

DOI:
https://doi.org/10.1090/S0002-9947-99-02183-2

Published electronically:
February 15, 1999

MathSciNet review:
1473442

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Abstract | References | Similar Articles | Additional Information

Abstract: An invariant inner product on the Lie algebra of a compact connected Lie group extends to a Hermitian inner product on the Lie algebra of the complexified Lie group . The Laplace-Beltrami operator, , on induced by the Hermitian inner product determines, for each number , a Green's function by means of the identity . The Hilbert space of holomorphic functions on which are square integrable with respect to is shown to be finite dimensional. It is spanned by the holomorphic extensions of the matrix elements of those irreducible representations of whose Casimir operator is appropriately related to .

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Additional Information

**Leonard Gross**

Affiliation:
Department of Mathematics, Cornell University, Ithaca, New York 14853

Email:
gross@math.cornell.edu

DOI:
https://doi.org/10.1090/S0002-9947-99-02183-2

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