K-invariant cusp forms for reductive symmetric spaces of split rank one
E.P. van den Ban, J.J. Kuit, H. Schlichtkrull, Forum Mathematicum 31 (2018) 341–349.
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Journal Article
| Published
| English
Author
van den Ban, Erik P.;
Kuit, Job J.;
Schlichtkrull, Henrik
Abstract
<jats:title>Abstract</jats:title>
<jats:p>Let <jats:inline-formula id="j_forum-2018-0150_ineq_9999_w2aab3b7c12b1b6b1aab1c17b1b1Aa">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>G</m:mi>
<m:mo>/</m:mo>
<m:mi>H</m:mi>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2018-0150_eq_0103.png" />
<jats:tex-math>{G/H}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula> be a reductive symmetric space of split rank one and let <jats:italic>K</jats:italic> be a maximal compact subgroup of <jats:italic>G</jats:italic>. In a previous article the first two authors introduced a notion of cusp forms for <jats:inline-formula id="j_forum-2018-0150_ineq_9998_w2aab3b7c12b1b6b1aab1c17b1b7Aa">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>G</m:mi>
<m:mo>/</m:mo>
<m:mi>H</m:mi>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2018-0150_eq_0103.png" />
<jats:tex-math>{G/H}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>. We show that the space of cusp forms coincides with the closure of the space of <jats:italic>K</jats:italic>-finite generalized matrix coefficients of discrete series representations if and only if there exist no <jats:italic>K</jats:italic>-spherical discrete series representations. Moreover, we prove that every <jats:italic>K</jats:italic>-spherical discrete series representation occurs with multiplicity one in the Plancherel decomposition of <jats:inline-formula id="j_forum-2018-0150_ineq_9997_w2aab3b7c12b1b6b1aab1c17b1c15Aa">
<jats:alternatives>
<m:math xmlns:m="http://www.w3.org/1998/Math/MathML">
<m:mrow>
<m:mi>G</m:mi>
<m:mo>/</m:mo>
<m:mi>H</m:mi>
</m:mrow>
</m:math>
<jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_forum-2018-0150_eq_0103.png" />
<jats:tex-math>{G/H}</jats:tex-math>
</jats:alternatives>
</jats:inline-formula>.</jats:p>
Publishing Year
Journal Title
Forum Mathematicum
Volume
31
Issue
2
Page
341-349
LibreCat-ID
Cite this
van den Ban EP, Kuit JJ, Schlichtkrull H. K-invariant cusp forms for reductive symmetric spaces of split rank one. Forum Mathematicum. 2018;31(2):341-349. doi:10.1515/forum-2018-0150
van den Ban, E. P., Kuit, J. J., & Schlichtkrull, H. (2018). K-invariant cusp forms for reductive symmetric spaces of split rank one. Forum Mathematicum, 31(2), 341–349. https://doi.org/10.1515/forum-2018-0150
@article{van den Ban_Kuit_Schlichtkrull_2018, title={K-invariant cusp forms for reductive symmetric spaces of split rank one}, volume={31}, DOI={10.1515/forum-2018-0150}, number={2}, journal={Forum Mathematicum}, publisher={Walter de Gruyter GmbH}, author={van den Ban, Erik P. and Kuit, Job J. and Schlichtkrull, Henrik}, year={2018}, pages={341–349} }
Ban, Erik P. van den, Job J. Kuit, and Henrik Schlichtkrull. “K-Invariant Cusp Forms for Reductive Symmetric Spaces of Split Rank One.” Forum Mathematicum 31, no. 2 (2018): 341–49. https://doi.org/10.1515/forum-2018-0150.
E. P. van den Ban, J. J. Kuit, and H. Schlichtkrull, “K-invariant cusp forms for reductive symmetric spaces of split rank one,” Forum Mathematicum, vol. 31, no. 2, pp. 341–349, 2018, doi: 10.1515/forum-2018-0150.
van den Ban, Erik P., et al. “K-Invariant Cusp Forms for Reductive Symmetric Spaces of Split Rank One.” Forum Mathematicum, vol. 31, no. 2, Walter de Gruyter GmbH, 2018, pp. 341–49, doi:10.1515/forum-2018-0150.
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