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Transactions of the American Mathematical Society

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Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups


Authors: L. Corwin, F. P. Greenleaf and G. Grélaud
Journal: Trans. Amer. Math. Soc. 304 (1987), 549-583
MSC: Primary 22E25; Secondary 22E27
DOI: https://doi.org/10.1090/S0002-9947-1987-0911085-6
MathSciNet review: 911085
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Abstract: Let $ K$ be a Lie subgroup of the connected, simply connected nilpotent Lie group $ G$, and let $ \mathfrak{k}$, $ \mathfrak{g}$ be the corresponding Lie algebras. Suppose that $ \sigma $ is an irreducible unitary representation of $ K$. We give an explicit direct integral decomposition of $ {\operatorname{Ind} _{k \to G}}\sigma $ into irreducibles. The description uses the Kirillov orbit picture, which gives a bijection between $ G^\wedge$ and the coadjoint orbits in $ {\mathfrak{g}^{\ast}}$ (and similarly for $ K^\wedge,\,{\mathfrak{k}^{\ast}}$). Let $ P:{\mathfrak{k}^{\ast}} \to {\mathfrak{g}^{\ast}}$ be the canonical projection, let $ {\mathcal{O}_\sigma } \subset {\mathfrak{k}^{\ast}}$ be the orbit corresponding to $ \sigma $, and, for $ \pi \in G^\wedge$, let $ {\mathcal{O}_\pi } \subset {\mathfrak{g}^{\ast}}$ be the corresponding orbit. The main result of the paper says essentially that $ \pi \in G^\wedge$ appears in the direct integral iff $ {P^{ - 1}}({\mathcal{O}_\sigma })$ meets $ {\mathcal{O}_\pi }$; the multiplicity of $ \pi $ is the number of $ {\operatorname{Ad} ^{\ast}}(K)$-orbits in $ {\mathcal{O}_\pi } \cap {P^{ - 1}}({\mathcal{O}_\sigma })$. There is also a natural description of the measure class in the integral.


References [Enhancements On Off] (What's this?)

  • [1] Y. Benoist, Espaces symmetriques exponentielles, Thesis III$ ^{me}$ cycle, Paris VII, 1983.
  • [2] I. K. Busiyatskaya, Representations of exponential Lie groups, J. Funct. Anal. Appl. 7 (1973), 151-152. (Russian) MR 0325855 (48:4201)
  • [3] L. Corwin, A representation-theoretic criterion for local solvability of left invariant differential operators on nilpotent Lie groups, Trans. Amer. Math. Soc. 264 (1981), 113-120. MR 597870 (83e:22013)
  • [4] L. Corwin and F. P. Greenleaf, Character formulas and spectra of compact nilmanifolds, J. Funct. Anal. 21 (1976), 123-154. MR 0393345 (52:14155)
  • [5] J. Fox, On the spectrum of compact nilmanifolds, preprint, 1984.
  • [6] E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Uspekhi Mat. Nauk 16 (1961), 93-119. MR 0131418 (24:A1269)
  • [7] G. Grelaud, Desintegration de representations induites d'un groupe de Lie resoluble exponentiel, C. R. Acad. Sci. Paris, Ser. A, 277 (1973), 327-330. MR 0325857 (48:4203)
  • [8] -, Sur les representations des groupes de Lie resoluble, Thesis III$ ^{me}$ cycle, Univ. de Poitiers, October, 1984.
  • [9] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspekhi Mat. Nauk 17 (1962), 57-110. MR 0142001 (25:5396)
  • [10] L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. Ecole Norm. Sup. 4 (1971), 457-608. MR 0439985 (55:12866)
  • [11] J. T. Schwartz, Differential geometry and topology, Gordon and Breach, New York, 1968.
  • [12] A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365-374. MR 0063994 (16:209a)
  • [13] H. Sussman, Analytic stratifications and subanalytic sets, monograph (in preparation).
  • [14] A. Tarski, A decision method for elementary algebra and geometry, 2nd ed., Univ. of California Press, Berkeley, 1951, 63 pp. MR 0044472 (13:423a)
  • [15] B. Van der Waerden, Modern algebra, 2nd ed., Ungar, New York, 1949.

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DOI: https://doi.org/10.1090/S0002-9947-1987-0911085-6
Article copyright: © Copyright 1987 American Mathematical Society

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