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
Full-text PDF
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
.
- [B] V. Bargmann, On a Hilbert space of analytic functions and an associated integral transform. Part I, Communications on Pure and Applied Mathematics 14 (1961), 187-214. MR 28:486
- [BO] C. Bender and S. Orszag, Advanced Mathematical Methods for Scientists and Engineers, McGraw Hill, 1978. MR 80d:00030
- [D] B. K. Driver, On the Kakutani-Itô-Segal-Gross and the Segal-Bargmann-Hall isomorphisms, J. of Funct. Anal. 133 (1995), 69-128. MR 97j:22020
- [DG] B. K. Driver and L. Gross, Hilbert spaces of holomorphic functions on complex Lie groups, Proceedings of the 1994 Taniguchi Symposium (to appear).
- [G] L. Gross, The homogeneous chaos over compact Lie groups in Stochastic Processes, A Festschrift in Honor of Gopinath Kallianpur, S. Cambanis et al., Eds., Springer-Verlag, New York, 1993, pp. 117-123. MR 97j:22021
- [H] B. Hall, The Segal-Bargmann ``coherent state'' transform for compact Lie groups, J. of Funct. Anal. 122 (1994), 103-151. MR 95e:22020
- [Hi] Omar Hijab, Hermite functions on compact Lie groups. I, J. of Funct. Anal. 125 (1994), 480-492. MR 96e:22017
- [K1] Paul Krée, Solutions faibles d'equations aux dérivées fonctionelles, Seminar Pierre Lelong I (1972/1973), in Lecture Notes in Mathematics, (See especially Sec.3), Vol. 410, Springer, New York/Berlin, 1974, pp.142-180. MR 51:8818
- [K2] Paul Krée, Solutions faibles d'equations aux dérivées fonctionelles, Seminar Pierre Lelong II (1973/1974), in Lecture Notes in Mathematics, (See especially Sec.5), Vol. 474, Springer, New York/Berlin, 1975, pp.16-47. MR 52:14998
- [K3] Paul Krée, Calcul d'intégrales et de dérivées en dimension infinie, J. of Funct. Anal. 31 (1979), 150-186. MR 80k:46051
- [S1] I. E. Segal, Tensor algebras over Hilbert spaces, Trans. Amer. Math. Soc. 81 (1956), 106-134. MR 17:880d
- [S2] I. E. Segal, Mathematical characterization of the physical vacuum for a linear Bose-Einstein field, Illinois J. Math. 6 (1962), 500-523. MR 26:1075
- [S3] I. E. Segal, The complex wave representation of the free Boson field, in ``Topics in functional analysis: essays dedicated to M. G. Krein on the occasion of his 70th birthday'', Advances in Mathematics: Supplementary studies, Vol. 3 (I. Gohberg and M. Kac, Eds.), Academic Press, New York, 1978, pp. 321-344. MR 82d:81069
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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