scholarly journals Invariant spherical distributions of discrete series on real semisimple symmetric spaces ${{G_C } \backslash {G_R }}$

1989 ◽  
Vol 65 (1) ◽  
pp. 12-14
Author(s):  
Shigeru Sano
Keyword(s):  
Symmetric Spaces ◽  
Discrete Series ◽  
Spherical Distributions

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

2019 ◽  
Vol 31 (2) ◽  
pp. 341-349
Author(s):  
Erik P. van den Ban ◽  
Job J. Kuit ◽  
Henrik Schlichtkrull
Keyword(s):  
Symmetric Spaces ◽  
Discrete Series ◽  
Series Representation ◽  
Compact Subgroup ◽  
Cusp Forms ◽  
Series Representations ◽  
Rank One ◽  
Discrete Series Representations ◽  
Generalized Matrix ◽  
Split Rank

AbstractLet {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}.

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