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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.
MSC:
 22E46 Semisimple Lie groups and their representations 22A20 Analysis on topological semigroups 22E40 Discrete subgroups of Lie groups
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