Complex algebraic geometry and calculation of multiplicities for induced representations of nilpotent Lie groups
HTML articles powered by AMS MathViewer
- by L. Corwin and F. P. Greenleaf PDF
- Trans. Amer. Math. Soc. 305 (1988), 601-622 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 algebraic-geometric 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
-
L. Corwin and F. P. Greenleaf, Unitary representations of nilpotent Lie groups and applications, Vol. 1, Cambridge Univ. Press, Cambridge, to appear in 1988.
- L. Corwin, F. P. Greenleaf, and G. Grélaud, Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups, Trans. Amer. Math. Soc. 304 (1987), no. 2, 549–583. MR 911085, DOI 10.1090/S0002-9947-1987-0911085-6
- Shigeru Iitaka, Algebraic geometry, North-Holland Mathematical Library, vol. 24, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties. MR 637060
- David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
- L. Pukánszky, Leçons sur les représentations des groupes, Monographies de la Société Mathématique de France, No. 2, Dunod, Paris, 1967 (French). MR 0217220
- L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) 4 (1971), 457–608. MR 439985 I. Shafarevich, Introduction to algebraic geometry, Grundlehren Math. Wiss., no. 213, Springer-Verlag, New York, 1975. H. Sussmann, Analytic stratifications and subanalytic sets (in preparation). B. Van der Waerden, Modern algebra, Vol. 2, 2nd ed., Ungar, New York, 1949.
- André Weil, Foundations of algebraic geometry, American Mathematical Society, Providence, R.I., 1962. MR 0144898
Additional Information
- © Copyright 1988 American Mathematical Society
- 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