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

  • Erik P. van den Ban , Job J. Kuit EMAIL logo and Henrik Schlichtkrull
Published/Copyright: October 30, 2018
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 2

Abstract

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.

Keywords: symmetric space; cusp form; discrete series representation
MSC 2010: 22E30; 22E45

Communicated by Karl-Hermann Neeb

Funding source: Deutsche Forschungsgemeinschaft

Award Identifier / Grant number: 262362164

Funding statement: The second author was supported by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 262362164.

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Received: 2018-06-21
Revised: 2018-08-30
Published Online: 2018-10-30
Published in Print: 2019-03-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
  3. Infinite families of equivariantly formal toric orbifolds
  4. Bounds for GL3L-functions in depth aspect
  5. Nonlinear Dirichlet problems with unilateral growth on the reaction
  6. K-invariant cusp forms for reductive symmetric spaces of split rank one
  7. Cellularity in quotient spaces of topological groups
  8. Shifted convolution sums for higher rank groups
  9. Dimension quotients, Fox subgroups and limits of functors
  10. Large sums of Hecke eigenvalues of holomorphic cusp forms
  11. K-theory classification of graded ultramatricial algebras with involution
  12. Regularity of symbolic powers and arboricity of matroids
  13. On some problems concerning symmetrization operators
  14. An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
  15. Rankin–Selberg gamma factors of level zero representations of GLn
  16. Sobolev’s inequality for double phase functionals with variable exponents
  17. Independence of Artin L-functions
  18. Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
https://doi.org/10.1515/forum-2018-0150
Keywords for this article
symmetric space; cusp form; discrete series representation

Articles in the same Issue

  1. Frontmatter
  2. Pseudo-differential operators on homogeneous spaces of compact and Hausdorff groups
  3. Infinite families of equivariantly formal toric orbifolds
  4. Bounds for GL3L-functions in depth aspect
  5. Nonlinear Dirichlet problems with unilateral growth on the reaction
  6. K-invariant cusp forms for reductive symmetric spaces of split rank one
  7. Cellularity in quotient spaces of topological groups
  8. Shifted convolution sums for higher rank groups
  9. Dimension quotients, Fox subgroups and limits of functors
  10. Large sums of Hecke eigenvalues of holomorphic cusp forms
  11. K-theory classification of graded ultramatricial algebras with involution
  12. Regularity of symbolic powers and arboricity of matroids
  13. On some problems concerning symmetrization operators
  14. An Erdős–Ko–Rado result for sets of pairwise non-opposite lines in finite classical polar spaces
  15. Rankin–Selberg gamma factors of level zero representations of GLn
  16. Sobolev’s inequality for double phase functionals with variable exponents
  17. Independence of Artin L-functions
  18. Oscillatory singular integral operators with Hölder class kernels on Hardy spaces
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