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The Plancherel formula for the horocycle spaces and generalizations, II

  • Ronald L. Lipsman (a1)

Abstract

The Plancherel formula for various semisimple homogeneous spaces with non-reductive stability group is derived within the framework of the Bonnet Plancherel formula for the direct integral decomposition of a quasi-regular representation. These formulas represent a continuation of the author's program to establish a new paradigm for concrete Plancherel analysis on homogeneous spaces wherein the distinction between finite and infinite multiplicity is de-emphasized. One interesting feature of the paper is the computation of the Bonnet nuclear operators corresponding to certain exponential representations (roughly those induced from infinite-dimensional representations of a subgroup). Another feature is a natural realization of the direct integral decomposition over a canonical set of concrete irreducible representations, rather than over the unitary dual.

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References

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[1]Baklouti, A. and Ludwig, J., ‘Opérateur d'entrelacement des représentations monomiales des groupes de Lie nilpotents’, to appear.
[2]Bonnet, P., ‘Transformation de Fourier des distributions de type positif sur un groupe de Lie unimodulaire’, J. Funct. Anal. 55 (1984), 220246.
[3]Duflo, M. and Moore, C., ‘On the regular representation of a non-unimodular locally compact group’, J. Funct. Anal. 21 (1976), 209243.
[4]Fujiwara, H., ‘Représentations monomiales des groupes de Lie nilpotents’, Pacific J. Math. 127 (1987), 329352.
[5]Fujiwara, H., ‘La formule de Plancherel pour les représentations monomiales des groupes de Lie nilpotents’, in: Représentation theory of Lie groups and Lie algebras (World Scientific, Singapore, 1992) pp. 140150.
[6]Goodman, R., ‘Complex Fourier analysis on nilpotent Lie groups’, Trans. Amer. Math. Soc. 160 (1971), 373391.
[7]Kleppner, A. and Lipsman, R., ‘The Plancherel formula for group extensions’, Ann. Scient. Ecole. Norm. Sup. 5 (1972), 459516
see also ‘The Plancherel formula for group extensions’, Ann. Scient. Ecole. Norm. Sup. 6 (1973), 103132.
[8]Kostant, B., ‘On the existence and irreducibility of certain series of representations’, in: Lie groups and their representations (Summer School of the Bolyai János Math. Soc.) (ed. B., Kostant) (Halsted Press, New York, 1975), pp. 231329.
[9]Lipsman, R., ‘The Penney-Fujiwara Plancherel formula for abelian symmetric spaces and completely solvable homogeneous spaces’, Pacific J. Math. 151 (1991), 265295.
[10]Lipsman, R., ‘The Plancherel formula for homogeneous spaces with polynomial spectrum’, Pacific J. Math. 159 (1993), 351377.
[11]Lipsman, R., ‘The Plancherel formula for the horocycle space and generalizations’, J. Austral. Math. Soc. (Ser. A) 61 (1996), 4256.
[12]Lipsman, R., ‘A unified approach to concrete Plancherel theory of homogeneous spaces’, Manuscripta Math. 94 (1997), 133149.
[13]Lipsman, R., ‘The Plancherel formula for homogeneous spaces with exponential spectrum’, J. Reine Ang. Math. (1998) to appear.
[14]Penney, R., ‘Abstract Plancherel theorems and a Frobenius reciprocity theorem’, J. Funct. Anal. 18 (1975), 177190.
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The Plancherel formula for the horocycle spaces and generalizations, II

  • Ronald L. Lipsman (a1)

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