K-invariant cusp forms for reductive symmetric spaces of split rank one

2018/06/21 by Ban, Erik P. van den, Kuit, Job J., Schlichtkrull, Henrik
#22E30 #22E45 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1806.08248

Abstract

Let G/H be a reductive symmetric space of split rank 1 and let K be a maximal compact subgroup of G. In a previous article the first two authors introduced a notion of cusp forms for G/H. We show that the space of cusp forms coincides with the closure of the K-finite generalized matrix coefficients of discrete series representations if and only if there exist no K-spherical discrete series representations. Moreover, we prove that every K-spherical discrete series representation occurs with multiplicity 1 in the Plancherel decomposition of G/H.

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