Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups
Authors:
L. Corwin and F. P. Greenleaf
Journal:
Trans. Amer. Math. Soc. 305 (1988), 601622
MSC:
Primary 22E27; Secondary 22E45
MathSciNet review:
924771
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Abstract: Let be a connected, simply connected nilpotent Lie group, a Lie subgroup, and an irreducible unitary representation of . In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of . One consequence of that work was that the multiplicity function was either a.e. infinite or a.e. bounded. In this paper, it is proved that if the multiplicity function is bounded, its parity is a.e. constant. The proof is algebraicgeometric in nature and amounts to an extension of the familiar fact that for almost all polynomials over of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if is a complex nilpotent Lie group and a complex Lie subgroup, then the multiplicity is a.e. constant.
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 D. Mumford, Algebraic geometry I: Complex projective varieties, Grundlehren Math. Wiss., no. 221, SpringerVerlag, Berlin and New York, 1976. MR 0453732 (56:11992)
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 H. Sussmann, Analytic stratifications and subanalytic sets (in preparation).
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 A. Weil, Foundations of algebraic geometry, Amer. Math. Soc. Colloq. Publ., vol. 29, Amer. Math. Soc., Providence, R.I., 1962. MR 0144898 (26:2439)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0002994719880924771X
PII:
S 00029947(1988)0924771X
Article copyright:
© Copyright 1988 American Mathematical Society
