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Transactions of the American Mathematical Society

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Canonical semi-invariants and the Plancherel formula for parabolic groups


Authors: Ronald L. Lipsman and Joseph A. Wolf
Journal: Trans. Amer. Math. Soc. 269 (1982), 111-131
MSC: Primary 22E30; Secondary 22E46, 43A80
DOI: https://doi.org/10.1090/S0002-9947-1982-0637031-6
MathSciNet review: 637031
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Abstract: A parabolic subgroup of a reductive Lie group is called "good" if the center of the universal enveloping algebra of its nilradical contains an element that is semi-invariant of weight proportional to the modular function. The "good" case is characterized here by invariance of the set of simple roots defining the parabolic, under the negative of the opposition element of the Weyl group. In the "good" case, the unbounded Dixmier-Pukanszky operator of the parabolic subgroup is described, the conditions under which it is a differential operator rather than just a pseudodifferential operator are specified, and an explicit Plancherel formula is derived for that parabolic.


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  • [1] M. Duflo, Sur les extensions des représentations des groupes de Lie nilpotents, Ann. Sci. École Norm. Sup. 5 (1972), 71-120. MR 0302823 (46:1966)
  • [2] A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977), 241-289. MR 0453829 (56:12082)
  • [3] F. Keene, Square integrable representations and a Plancherel theorem for parabolic groups, Trans. Amer. Math. Soc. 243 (1978), 61-73. MR 0498983 (58:16967)
  • [4] F. Keene, R. Lipsman, and J. A. Wolf, The Plancherel formula for parabolic subgroups, Israel J. Math. 28 (1977), 68-90. MR 0507242 (58:22400)
  • [5] A. Kleppner and R. Lipsman, The Plancherel formula for group extensions, Ann. Sci. École Norm. Sup. 5 (1972), 459-516. MR 0342641 (49:7387)
  • [6] R. Lipsman, Fourier inversion on Borel subgroups of Chevalley groups, Trans. Amer. Math. Soc. 260 (1980), 607-622. MR 574803 (81g:22022)
  • [7] R. Lipsman and J. A. Wolf, The Plancherel formula for parabolic subgroups of the classical groups, J. Analyse Math. 34 (1978), 120-161. MR 531273 (81f:22022)
  • [8] L. Pukanszky, Representations of solvable Lie groups, Ann. Sci. École Norm. Sup. 4 (1971), 464-608. MR 0439985 (55:12866)
  • [9] M. Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ciênc. 35 (1963), 487-489. MR 0171782 (30:2009)
  • [10] J. A. Wolf, The action of a real semisimple group on a complex flag manifold. I, Bull. Amer. Math. Soc. 75 (1969), 1121-1237. MR 0251246 (40:4477)
  • [11] -, Unitary representations of maximal parabolic subgroups of the classical groups, Mem. Amer. Math. Soc. no. 180 (1976). MR 0444847 (56:3194)
  • [12] -, Classification and Fourier inversion for parabolic subgroups with square integrable nilradical, Mem. Amer. Math. Soc. no. 225 (1979). MR 546511 (81a:22010)
  • [13] J. Dixmier, Sur les représentations unitaires des groupes de Lie nilpotent. IV, Canad. J. Math. 11 (1959), 321-344. MR 0106963 (21:5693)
  • [14] M. Raïs, La représentation coadjointe du groupe affine, Ann. Inst. Fourier (Grenoble) 38 (1978), 207-237. MR 500922 (81c:17017)

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DOI: https://doi.org/10.1090/S0002-9947-1982-0637031-6
Article copyright: © Copyright 1982 American Mathematical Society

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