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A Plancherel formula for idyllic nilpotent Lie groups


Author: Eloise Carlton
Journal: Trans. Amer. Math. Soc. 224 (1976), 1-42
MSC: Primary 22E25
MathSciNet review: 0425014
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Abstract: A procedure is developed which can be used to compute the Plancherel measure for a certain class of nilpotent Lie groups, including the Heisenberg groups, free groups, two-and three-step groups, the nilpotent part of an Iwasawa decomposition of the R-split form of the classical simple groups $ {A_l},{C_l},{G_2}$.

Let G be a connected, simply connected nilpotent Lie group. The Plancherel formula for G can be expressed in terms of Plancherel measure of a normal subgroup N and projective Plancherel measures of certain subgroups of $ G/N$. To get an explicit measure for G, we need an explicit formula for (1) the disintegration of Plancherel measure of N under the action of G on N, and (2) projective Plancherel measures of $ {G_\gamma }/N$, where $ {G_\gamma }$ is the stability subgroup at $ \gamma $ in N. When both N and $ {G_\gamma }/N$ are abelian, the measures (1) and (2) are obtained as special cases of more general problems. These measures combine into Plancherel measure for G.


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  • [1] Larry Baggett and Adam Kleppner, Multiplier representations of abelian groups, J. Functional Analysis 14 (1973), 299–324. MR 0364537
  • [2] P. Bernat, N. Conze, M. Duflo, M. Lévy-Nahas, M. Rais, P. Renouard and M. Vergne, Représentations des groupes de Lie résolubles, Dunod, Paris, 1972.
  • [3] N. Bourbaki, Éléments de mathématique. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre 9: Formes sesquilinéaires et formes quadratiques, Actualités Sci. Ind. no. 1272, Hermann, Paris, 1959 (French). MR 0107661
  • [4] N. Bourbaki, Éléments de mathématique. Fasc. XXXVII. Groupes et algèbres de Lie. Chapitre II: Algèbres de Lie libres. Chapitre III: Groupes de Lie, Hermann, Paris, 1972. Actualités Scientifiques et Industrielles, No. 1349. MR 0573068
  • [5] -, Intégration. Chap. 5: Intégration des mesures, Actualités Sci. Indust., no. 1244, Hermann, Paris, 1967. MR 35 #322.
  • [6] -, Intégration. Chap. 6: Intégration vectorielle, Hermann, Paris, 1959. MR 23 #A2033.
  • [7] Jacques Dixmier, Les 𝐶*-algèbres et leurs représentations, Deuxième édition. Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars Éditeur, Paris, 1969 (French). MR 0246136
  • [8] J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. II, Bull. Soc. Math. France 85 (1957), 325–388 (French). MR 0095426
  • [9] Jacques Dixmier, Sur les représentations unitaries des groupes de Lie nilpotents. III, Canad. J. Math. 10 (1958), 321–348. MR 0095427
  • [10] Hans Freudenthal and H. de Vries, Linear Lie groups, Pure and Applied Mathematics, Vol. 35, Academic Press, New York-London, 1969. MR 0260926
  • [11] Melvin Hausner and Jacob T. Schwartz, Lie groups; Lie algebras, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0235065
  • [12] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
  • [13] A. A. Kirillov, Plancherel’s measure for nilpotent Lie groups, Funkcional. Anal. i Priložen 1 (1967), no. 4, 84–85 (Russian). MR 0224748
  • [14] Adam Kleppner and Ronald L. Lipsman, The Plancherel formula for group extensions. I, II, Ann. Sci. École Norm. Sup. (4) 5 (1972), 459–516; ibid. (4) 6 (1973), 103–132. MR 0342641
  • [15] -, The Plancherel formula for group extensions. II, Ann. Sci. École Norm. Sup. (4) 6 (1973), 103-132. MR 49 #7387.
  • [16] George W. Mackey, Induced representations of locally compact groups. I, Ann. of Math. (2) 55 (1952), 101–139. MR 0044536
  • [17] George W. Mackey, Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311. MR 0098328
  • [18] M. Plancherel, Contribution à l'étude de la représentation d'une fonction arbitraire par des intégrales définies, Rend. Circ. Mat. Palermo 30 (1910), 289-335.
  • [19] L. Pukańszky, Leçons sur les représentations des groupes, Monographies Soc. Math. France, no. 2, Dunod, Paris, 1967. MR 36 #311.
  • [20] Michael Spivak, Calculus on manifolds. A modern approach to classical theorems of advanced calculus, W. A. Benjamin, Inc., New York-Amsterdam, 1965. MR 0209411
  • [21] Garth Warner, Harmonic analysis on semi-simple Lie groups. I, Springer-Verlag, New York-Heidelberg, 1972. Die Grundlehren der mathematischen Wissenschaften, Band 188. MR 0498999
  • [22] André Weil, L’intégration dans les groupes topologiques et ses applications, Actual. Sci. Ind., no. 869, Hermann et Cie., Paris, 1940 (French). [This book has been republished by the author at Princeton, N. J., 1941.]. MR 0005741
  • [23] E. Carlton, A Plancherel formula for some nilpotent Lie groups, Ph. D. Thesis, University of Colorado, 1974.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0425014-8
Keywords: Connected, simply connected real nilpotent Lie group, nilpotent Lie algebra, Jordan-Hölder basis, unipotent action, contragredient action, G-invariant measure, Zariski open set, disintegration of measures, idyll, coadjoint representation, stability subgroup, multipler representation, irreducible representation, projective Plancherel measure, Plancherel formula for group extensions
Article copyright: © Copyright 1976 American Mathematical Society