Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups
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 by L. Corwin and F. P. Greenleaf PDF
 Trans. Amer. Math. Soc. 305 (1988), 601622 Request permission
Abstract:
Let $G$ be a connected, simply connected nilpotent Lie group, $H$ a Lie subgroup, and $\sigma$ an irreducible unitary representation of $H$. In a previous paper, the authors and G. Grelaud gave an explicit direct integral decomposition (with multiplicities) of $\operatorname {Ind} (H \uparrow G, \sigma )$. 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 $R$ of fixed degree, the parity of the number of roots is a.e. constant. One consequence of the methods is that if $G$ is a complex nilpotent Lie group and $H$ a complex Lie subgroup, then the multiplicity is a.e. constant.References

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Additional Information
 © Copyright 1988 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 305 (1988), 601622
 MSC: Primary 22E27; Secondary 22E45
 DOI: https://doi.org/10.1090/S0002994719880924771X
 MathSciNet review: 924771