Wave packets for the relative discrete series. II: The nonholomorphic case (Q579440)

This paper continues the investigations of the authors in the harmonic analysis on semisimple Lie groups without the finiteness condition for the center [see, for example, Compos. Math. 57, 271-355 (1986; Zbl 0587.22005)]. Let G be a connected simple Lie group with infinite center, K be a maximal compactly embedded subgroup. (Then G is simply connected and G/K is a Hermitian symmetric space.) In this paper wave packets of K- finite matrix elements of relative discrete series representations are constructed by the integration on bounded intervals which parametrize this series [the case of unbounded intervals is considered in the first part of this work (see the preceding review)]. The main result of the paper is that linear combinations of all these wave packets form a dense subspace of the space of cusp forms. This is the analogue of Harish-Chandra's result for groups with finite center. On the other hand, it is proved that one cannot expect a direct analogue of Harish-Chandra's theorem about the decomposition of the Schwartz space.