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), 601-622

MSC:
Primary 22E27; Secondary 22E45

DOI:
https://doi.org/10.1090/S0002-9947-1988-0924771-X

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 algebraic-geometric 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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DOI:
https://doi.org/10.1090/S0002-9947-1988-0924771-X

Article copyright:
© Copyright 1988
American Mathematical Society