Canonical semi-invariants and the Plancherel formula for parabolic groups

Lipsman, Ronald L.; Wolf, Joseph A.

doi:10.1090/S0002-9947-1982-0637031-6

Canonical semi-invariants and the Plancherel formula for parabolic groups
HTML articles powered by AMS MathViewer

by Ronald L. Lipsman and Joseph A. Wolf

Trans. Amer. Math. Soc. 269 (1982), 111-131

DOI: https://doi.org/10.1090/S0002-9947-1982-0637031-6

PDF | Request permission

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.

References

Michel Duflo, Sur les extensions des représentations irréductibles des groupes de Lie nilpotents, Ann. Sci. École Norm. Sup. (4) 5 (1972), 71–120 (French). MR 302823
A. Joseph, A preparation theorem for the prime spectrum of a semisimple Lie algebra, J. Algebra 48 (1977), no. 2, 241–289. MR 453829, DOI 10.1016/0021-8693(77)90306-4
Frederick W. Keene, Square integrable representations and a Plancherel theorem for parabolic subgroups, Trans. Amer. Math. Soc. 243 (1978), 61–73. MR 498983, DOI 10.1090/S0002-9947-1978-0498983-X
Frederick W. Keene, Ronald L. Lipsman, and Joseph A. Wolf, The Plancherel formula for parabolic subgroups, Israel J. Math. 28 (1977), no. 1-2, 68–90. MR 507242, DOI 10.1007/BF02759782
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 342641
Ronald L. Lipsman, Fourier inversion on Borel subgroups of Chevalley groups: the symplectic group case, Trans. Amer. Math. Soc. 260 (1980), no. 2, 607–622. MR 574803, DOI 10.1090/S0002-9947-1980-0574803-9
Ronald L. Lipsman and Joseph A. Wolf, The Plancherel formula for parabolic subgroups of the classical groups, J. Analyse Math. 34 (1978), 120–161 (1979). MR 531273, DOI 10.1007/BF02790010
L. Pukanszky, Unitary representations of solvable Lie groups, Ann. Sci. École Norm. Sup. (4) 4 (1971), 457–608. MR 439985
Maxwell Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci. 35 (1963), 487–489. MR 171782
Joseph A. Wolf, The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components, Bull. Amer. Math. Soc. 75 (1969), 1121–1237. MR 251246, DOI 10.1090/S0002-9904-1969-12359-1
Joseph A. Wolf, Unitary representations of maximal parabolic subgroups of the classical groups, Mem. Amer. Math. Soc. 8 (1976), no. 180, iii+193. MR 444847, DOI 10.1090/memo/0180
Joseph A. Wolf, Classification and Fourier inversion for parabolic subgroups with square integrable nilradical, Mem. Amer. Math. Soc. 22 (1979), no. 225, iii+166. MR 546511, DOI 10.1090/memo/0225
Jacques Dixmier, Sur les représentations unitaires des groupes de Lie nilpotents. IV, Canadian J. Math. 11 (1959), 321–344 (French). MR 106963, DOI 10.4153/CJM-1959-034-8
Mustapha Raïs, La représentation coadjointe du groupe affine, Ann. Inst. Fourier (Grenoble) 28 (1978), no. 1, xi, 207–237 (French, with English summary). MR 500922