The Penney-Fujiwara Plancherel formula for Gelfand pairs. (English) Zbl 0866.22014
Let \(G=H\ltimes N\) be a semidirect product, where \(H\) is compact and \(N\) is simply connected nilpotent. The author studied in [J. Math. Pures Appl., IX. Ser. 59, 337-374 (1980; Zbl 0441.22007)] the harmonic analysis of such a group by using the orbit method and the Mackey machinery. This paper is concerned with the quasi-regular representation \(\tau\) of the homogeneous space \(G/H\). When \((G,H)\) is a Gelfand pair, an explicit description is given for the Penney’s distributions to get a concrete form of Penney’s Plancherel formula for \(\tau\) [R. Penney, J. Funct. Anal. 18, 177-190 (1975; Zbl 0305.22016)]. We must use complex polarizations but the Penney’s distributions take exactly the same form as in the case of real polarizations. One of the other main results is a distribution-theoretic form of Frobenius reciprocity for Gelfand pairs.
The author has been pursuing an extensive research on the subject [see, e.g., Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012), etc.]; some of the results of this paper were announced in [Representation theory of Lie groups and Lie algebras, World Scientific Press, Singapore, 120-139 (1992)].
The author has been pursuing an extensive research on the subject [see, e.g., Pac. J. Math. 151, 265-295 (1991; Zbl 0759.22012), etc.]; some of the results of this paper were announced in [Representation theory of Lie groups and Lie algebras, World Scientific Press, Singapore, 120-139 (1992)].
Reviewer: H.Fujiwara (Iizuka)
MSC:
| 22E45 | Representations of Lie and linear algebraic groups over real fields: analytic methods |
Keywords:
harmonic analysis; orbit method; quasi-regular representation; homogeneous space; Gelfand pair; Penney’s distributions; Penney’s Plancherel formulaReferences:
| [1] | N. Anh, Restriction of the principal series of \(SL(n,C)\) to some reductive subgroups , Pacific J. Math. 38 (1971), 295-313. · Zbl 0198.05003 · doi:10.2140/pjm.1971.38.295 |
| [2] | C. Benson, J. Jenkins, R. Lipsman and G. Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group , Pacific J. Math. (1996), · Zbl 0868.22015 · doi:10.2140/pjm.1997.178.1 |
| [3] | C. Benson, J. Jenkins and G. Ratcliff, On Gelfand pairs associated with solvable Lie groups , Trans. Amer. Math. Soc. 321 (1990), 85-116. JSTOR: · Zbl 0704.22006 · doi:10.2307/2001592 |
| [4] | L. Corwin, F. Greenleaf and G. Grélaud, Direct integral decompositions and multiplicities for induced representations of nilpotent Lie groups , Trans. Amer. Math. Soc. 304 (1987), 549-583. JSTOR: · Zbl 0629.22005 · doi:10.2307/2000731 |
| [5] | J. Dixmier, Les \(C^*\)-algèbres et leurs représentations , Gauthier-Villars, Paris, 1964. · Zbl 0152.32902 |
| [6] | M. Duflo, Sur les extensions des représentations irréductibles des groupes de Lie nilpotents , Ann. Scient. École Norm. Sup. 5 (1972), 71-120. · Zbl 0241.22030 |
| [7] | H. Fujiwara, Représentations monomiales d’un groupe de Lie nilpotente , Pacific J. Math. 127 (1987), 329-335. · Zbl 0588.22008 |
| [8] | G. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups , Inventiones Mathematicae 67 (1982), 333-356. · Zbl 0497.22006 · doi:10.1007/BF01393821 |
| [9] | S. Helgason, Groups and geometric analysis , Academic Press, New York, 1984. · Zbl 0543.58001 |
| [10] | R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions , Math. Ann. 290 (1991), 565-619. · Zbl 0733.20019 · doi:10.1007/BF01459261 |
| [11] | R. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical , J. Math. Pures et Appl. 59 (1980), 337-374. · Zbl 0421.22005 |
| [12] | ——–, Harmonic analysis on non-semisimple symmetric spaces , Israel J. Math. 54 (1986), 335-350. · Zbl 0611.43002 · doi:10.1007/BF02764962 |
| [13] | ——–, Orbital parameters for induced and restricted representations , Trans. Amer. Math. Soc. 313 (1989), 433-473. · Zbl 0683.22009 · doi:10.2307/2001416 |
| [14] | ——–, The Penney-Fujiwara Plancherel formula for symmetric spaces , in Progress in math. : The orbit method in representation theory , Birkhaüser 82 (1990), 135-145. |
| [15] | ——–, The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces , Pacific J. Math. 151 (1991), 265-295. · Zbl 0759.22012 · doi:10.2140/pjm.1991.151.265 |
| [16] | ——–, The Plancherel formula for homogeneous spaces with polynomial spectrum , Pacific J. Math. 159 (1993), 351-377. · Zbl 0798.22005 · doi:10.2140/pjm.1993.159.351 |
| [17] | ——–, The Penney-Fujiwara Plancherel formula for homogeneous spaces , in Representation theory of Lie groups and Lie algebras : Proceedings of Fuji-Kawaguchiko conference , World Scientific Press, Singapore, 1992, 120-139. |
| [18] | T. Nomura, Harmonic analysis on a nilpotent Lie group and representations of a solvable Lie group on \(\bar\partial_b\) cohomology spaces , Japanese J. Math. 13 (1987), 277-332. · Zbl 0646.22005 |
| [19] | R. Penney, Abstract Plancherel theorems and a Frobenius reciprocity theorem , J. Funct. Anal. 18 (1975), 177-190. · Zbl 0305.22016 · doi:10.1016/0022-1236(75)90023-3 |
| [20] | N. Poulsen, On \(C^\infty\)-vectors and intertwining bilinear forms for representations of Lie groups , J. Funct. Anal. 9 (1972), 87-120. · Zbl 0237.22013 · doi:10.1016/0022-1236(72)90016-X |
| [21] | N. Wildberger, The moment map of a Lie group representation , Trans. Amer. Math. Soc. 330 (1992), 257-268. JSTOR: · Zbl 0762.22012 · doi:10.2307/2154163 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
