Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups
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 by L. Corwin, F. P. Greenleaf and G. Grélaud PDF
 Trans. Amer. Math. Soc. 304 (1987), 549583 Request permission
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

Y. Benoist, Espaces symmetriques exponentielles, Thesis III$^{me}$ cycle, Paris VII, 1983.
 I. K. Busjackaja, The representations of exponential Lie groups, Funkcional. Anal. i Priložen. 7 (1973), no. 2, 79–80 (Russian). MR 0325855
 Lawrence Corwin, A representationtheoretic criterion for local solvability of left invariant differential operators on nilpotent Lie groups, Trans. Amer. Math. Soc. 264 (1981), no. 1, 113–120. MR 597870, DOI 10.1090/S00029947198105978706
 Lawrence Corwin and Frederick P. Greenleaf, Character formulas and spectra of compact nilmanifolds, J. Functional Analysis 21 (1976), no. 2, 123–154. MR 0393345, DOI 10.1016/00221236(76)900744 J. Fox, On the spectrum of compact nilmanifolds, preprint, 1984.
 E. A. Gorin, Asymptotic properties of polynomials and algebraic functions of several variables, Uspehi Mat. Nauk 16 (1961), no. 1 (97), 91–118 (Russian). MR 0131418
 Gérard Grélaud, Désintégration de représentations induites d’un groupe de Lie résoluble exponentiel, C. R. Acad. Sci. Paris Sér. AB 277 (1973), A327–A330 (French). MR 325857 —, Sur les representations des groupes de Lie resoluble, Thesis III$^{me}$ cycle, Univ. de Poitiers, October, 1984.
 A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
 L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) 4 (1971), 457–608. MR 439985 J. T. Schwartz, Differential geometry and topology, Gordon and Breach, New York, 1968.
 A. Seidenberg, A new decision method for elementary algebra, Ann. of Math. (2) 60 (1954), 365–374. MR 63994, DOI 10.2307/1969640 H. Sussman, Analytic stratifications and subanalytic sets, monograph (in preparation).
 Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, BerkeleyLos Angeles, Calif., 1951. 2nd ed. MR 0044472 B. Van der Waerden, Modern algebra, 2nd ed., Ungar, New York, 1949.
Additional Information
 © Copyright 1987 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 304 (1987), 549583
 MSC: Primary 22E25; Secondary 22E27
 DOI: https://doi.org/10.1090/S00029947198709110856
 MathSciNet review: 911085