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

MathSciNet review:
924771

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

**[1]**L. Corwin and F. P. Greenleaf,*Unitary representations of nilpotent Lie groups and applications*, Vol. 1, Cambridge Univ. Press, Cambridge, to appear in 1988.**[2]**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**, 10.1090/S0002-9947-1987-0911085-6**[3]**Shigeru Iitaka,*Algebraic geometry*, Graduate Texts in Mathematics, vol. 76, Springer-Verlag, New York-Berlin, 1982. An introduction to birational geometry of algebraic varieties; North-Holland Mathematical Library, 24. MR**637060****[4]**David Mumford,*Algebraic geometry. I*, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties; Grundlehren der Mathematischen Wissenschaften, No. 221. MR**0453732****[5]**L. Pukánszky,*Leçons sur les représentations des groupes*, Monographies de la Société Mathématique de France, No. 2, Dunod, Paris, 1967 (French). MR**0217220****[6]**L. Pukanszky,*Unitary representations of solvable Lie groups*, Ann. Sci. École Norm. Sup. (4)**4**(1971), 457–608. MR**0439985****[7]**I. Shafarevich,*Introduction to algebraic geometry*, Grundlehren Math. Wiss., no. 213, Springer-Verlag, New York, 1975.**[8]**H. Sussmann,*Analytic stratifications and subanalytic sets*(in preparation).**[9]**B. Van der Waerden,*Modern algebra*, Vol. 2, 2nd ed., Ungar, New York, 1949.**[10]**André Weil,*Foundations of algebraic geometry*, American Mathematical Society, Providence, R.I., 1962. MR**0144898**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
22E27,
22E45

Retrieve articles in all journals with MSC: 22E27, 22E45

Additional Information

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

Article copyright:
© Copyright 1988
American Mathematical Society