Asymptotic behaviour of matrix coefficients related to reductive symmetric spaces. (English) Zbl 0629.43008

van den Ban, E. P.

Indag. Math. 49, 225-249 (1987).

Let \(X=G/H\) be a reductive symmetric space, with G of Harish-Chandra’s class, \(\tau\) an involution of G, and H an open subgroup of the group of fixed points by \(\tau\) ; let K be a \(\tau\)-stable maximal compact subgroup of G.
The author studies the asymptotic behaviour of K-finite functions on X which are annihilated by a cofinite ideal of the center Z of the enveloping algebra of G. Methods and results of Harish-Chandra and Casselman-Miličić for the group case extend to the present situation, with some differences.
The results are applied to study the space of Z- and K-finite Schwartz functions on X.

Reviewer: F.Rouvière

MSC:

43A85	Harmonic analysis on homogeneous spaces
22E46	Semisimple Lie groups and their representations

Keywords:

matrix coefficients of representations; reductive symmetric space; asymptotic behaviour; enveloping algebra; Schwartz functions