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Journal of Lie Theory 23 (2013), No. 2, 433--458
Copyright Heldermann Verlag 2013

Multiplicity Free Spaces with a One-Dimensional Quotient

Hubert Rubenthaler
Institut de Recherche Mathématique Avancée, Université de Strasbourg et CNRS, 7 rue René Descartes, 67084 Strasbourg, France

The multiplicity free spaces with a one dimensional quotient were introduced by Thierry Levasseur ["Radial components, prehomogeneous vector spaces, and rational Cherednik algebras", Int. Math. Res. Not. IMRN 2009, no 3, 462--511]. Recently, the author has shown that the algebra of differential operators on such spaces which are invariant under the semi-simple part of the group is a Smith algebra ["Invariant differential operators on a class of multiplicity free spaces", arXiv:1103.1721v1 (math.RT)]. We give here the classification of these spaces which are indecomposable, up to geometric equivalence. We also investigate whether or not these spaces are regular or of parabolic type as a prehomogeneous vector space.

Keywords: Multiplicity free spaces, one dimensional quotient, prehomogeneous vector spaces.

MSC: 14L30, 11S90, 22E46

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