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Vanishing properties of analytically continued matrix coefficients. (English) Zbl 1012.22027
The authors consider generalized matrix coefficients of unitary highest weight representations of a linear hermitian group \(G\). By earlier results of Olshanski and Stanton these analytically extend to a complex Olshanski semigroup \(S\), where \(G\subseteq S\subseteq G_{{\mathbb C}}\). A principal result of the paper under review is a vanishing theorem of the Howe-Moore type for the analytically extended matrix coefficients, thus extending the Howe-Moore theorem from \(G\) to \(S\) for the special class of highest weight representations. The second principal result is a similar vanishing theorem for cusp forms.
22E46 Semisimple Lie groups and their representations
22A20 Analysis on topological semigroups
22E40 Discrete subgroups of Lie groups
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