Multiplicity-free representations of certain nilpotent Lie groups over Siegel domains of the second kind

Abstract

We investigate the multiplicity-freeness property for the holomorphic multiplier representations of affine transformation groups of a Siegel domain of the second kind. We deal with the generalized Heisenberg group and its subgroups. Necessary and sufficient conditions for a specific representation to be multiplicity-free are provided. We study the multiplicity-freeness property in relation to the geometrical notions of coisotropic action and visible action, and also the commutativity of the algebra of invariant differential operators.

17 pages. /v2: 22 pages. /v3: Remark 2.2 added, minor mistakes corrected, fully revised version will appear in the Proceedings of the 7th Tunisian-Japanese Conference, Springer Proceedings in Mathematics & Statistics