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

TY - JOUR

T1 - K-invariant cusp forms for reductive symmetric spaces of split rank one

AU - Ban van den, Erik

AU - Kuit, Job Jacob

AU - Schlichtkrull, Henrik

PY - 2019/3/1

Y1 - 2019/3/1

N2 - Let G/H be a reductive symmetric space of split rank one 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 space of 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 one in the Plancherel decomposition of G/H.

AB - Let G/H be a reductive symmetric space of split rank one 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 space of 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 one in the Plancherel decomposition of G/H.

U2 - 10.1515/forum-2018-0150

DO - 10.1515/forum-2018-0150

M3 - Journal article

SN - 0933-7741

VL - 31

SP - 341

EP - 349

JO - Forum Mathematicum

JF - Forum Mathematicum

IS - 2

ER -