Rocky Mountain Journal of Mathematics

The Penney-Fujiwara Plancherel Formula for Gelfand Pairs

Ronald L. Lipsman

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Rocky Mountain J. Math., Volume 26, Number 2 (1996), 655-677.

First available in Project Euclid: 5 June 2007

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Zentralblatt MATH identifier

Primary: 22E45: Representations of Lie and linear algebraic groups over real fields: analytic methods {For the purely algebraic theory, see 20G05}


Lipsman, Ronald L. The Penney-Fujiwara Plancherel Formula for Gelfand Pairs. Rocky Mountain J. Math. 26 (1996), no. 2, 655--677. doi:10.1216/rmjm/1181072078.

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  • N. Anh, Restriction of the principal series of $SL(n,C)$ to some reductive subgroups, Pacific J. Math. 38 (1971), 295-313.
  • C. Benson, J. Jenkins, R. Lipsman and G. Ratcliff, A geometric criterion for Gelfand pairs associated with the Heisenberg group, Pacific J. Math. (1996), to appear.
  • C. Benson, J. Jenkins and G. Ratcliff, On Gelfand pairs associated with solvable Lie groups, Trans. Amer. Math. Soc. 321 (1990), 85-116.
  • 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.
  • J. Dixmier, Les $C^*$-algèbres et leurs représentations, Gauthier-Villars, Paris, 1964.
  • M. Duflo, Sur les extensions des représentations irréductibles des groupes de Lie nilpotents, Ann. Scient. École Norm. Sup. 5 (1972), 71-120.
  • H. Fujiwara, Représentations monomiales d'un groupe de Lie nilpotente, Pacific J. Math. 127 (1987), 329-335.
  • G. Heckman, Projections of orbits and asymptotic behavior of multiplicities for compact connected Lie groups, Inventiones Mathematicae 67 (1982), 333-356.
  • S. Helgason, Groups and geometric analysis, Academic Press, New York, 1984.
  • R. Howe and T. Umeda, The Capelli identity, the double commutant theorem, and multiplicity-free actions, Math. Ann. 290 (1991), 565-619.
  • R. Lipsman, Orbit theory and harmonic analysis on Lie groups with co-compact nilradical, J. Math. Pures et Appl. 59 (1980), 337-374.
  • --------, Harmonic analysis on non-semisimple symmetric spaces, Israel J. Math. 54 (1986), 335-350.
  • --------, Orbital parameters for induced and restricted representations, Trans. Amer. Math. Soc. 313 (1989), 433-473.
  • --------, The Penney-Fujiwara Plancherel formula for symmetric spaces, in Progress in math.: The orbit method in representation theory, Birkhaüser 82 (1990), 135-145.
  • --------, The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces, Pacific J. Math. 151 (1991), 265-295.
  • --------, The Plancherel formula for homogeneous spaces with polynomial spectrum, Pacific J. Math. 159 (1993), 351-377.
  • --------, 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.
  • 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.
  • R. Penney, Abstract Plancherel theorems and a Frobenius reciprocity theorem, J. Funct. Anal. 18 (1975), 177-190.
  • N. Poulsen, On $C^\infty$-vectors and intertwining bilinear forms for representations of Lie groups, J. Funct. Anal. 9 (1972), 87-120.
  • N. Wildberger, The moment map of a Lie group representation, Trans. Amer. Math. Soc. 330 (1992), 257-268.