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Let G be a semisimple Lie group with a finite number of connected components and a finite center. Let K be a maximal compact subgroup. Let X be a smooth G-space equipped with a G-invariant measure. In this paper, we give upper bounds for K-finite and -smooth matrix coefficients of the regular representation L2(X) under an assumption about supp. Furthermore, we show that this bound holds for unitary representations that are weakly contained in L2(X). Our result generalizes a result of Cowling-Haagerup-Howe (J Reine Angew Math 387:97-110, 1988). As an example, we discuss the matrix coefficients of the O(p, q) representation. © 2009 Birkhäuser Verlag Basel/Switzerland.

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Selecta Mathematica, New Series

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