Abstract
Let K be a compact Lie group acting by automorphisms on a nilpotent Lie group N. One calls (K, N) a Gelfand pair when the integrable K-invariant functions on N form a commutative algebra under convolution. We prove that in this case the coadjoint orbits for G:= K × N which meet the annihilator
of the Lie algebra
of K do so in single K-orbits. This generalizes a result of the authors and R. Lipsman concerning Gelfand pairs associated with Heisenberg groups.
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Benson, C., Jenkins, J. & Ratcliff, G. The orbit method and Gelfand pairs, associated with nilpotent Lie groups. J Geom Anal 9, 569–582 (1999). https://doi.org/10.1007/BF02921973
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DOI: https://doi.org/10.1007/BF02921973