Conditions for the nontriviality of the Hilbert space of a holomorphically induced representation of a solvable Lie group

The notion of holomorphically induced representation is ageneralization of the concept of representation induced by a representation of a subgroup. It permitted Kostant and Auslander to give a classification of irreducible unitary representations of solvable Lie groups. A holomorphically induced representation is constructed in a function space on the group, where the functions satisfy a number of algebraic conditions and lie in some $L^2$-space. It may happen that some nontrivial functions satisfy the algebraic conditions but none of them lie in $L^2$. In this paper a necessary and sufficient condition that this not occur when the Lie group under consideration is solvable is proved. The condition involves the Lie algebra and the parameters appearing in the definition of the representation.
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