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In this chapter, we consider the unitary symmetric spaces of the form \(X = U(p,q)/U(1)U(p-1,q)\) and their discrete series representations. Inspired by the work of A. Venkatesh and Y. Sakellaridis on L-groups of p-adic spherical spaces Sakellaridis and Venkatesh (Asterisque 396, 2017) , we formulate and prove natural relative branching laws for the restriction of such discrete series representations to smaller groups of the same type and corresponding unitary symmetric spaces. Using period integrals, some results on the Laplacian of spheres, Schlichtkrull et al. (Sao Paulo J Math Sci 12(2), 295–358, 2018) and results by Kobayashi (Adv Math 388, 38, 2021), we prove an analogue of these conjectures for rank-one unitary symmetric spaces.

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Toward Gan-Gross-Prasad-Type Conjecture for Discrete Series Representations of Symmetric Spaces

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  • 发表日期：

    2025年

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  • Bent Ørsted,
  • Birgit Speh

摘要

In this chapter, we consider the unitary symmetric spaces of the form \(X = U(p,q)/U(1)U(p-1,q)\) and their discrete series representations. Inspired by the work of A. Venkatesh and Y. Sakellaridis on L-groups of p-adic spherical spaces Sakellaridis and Venkatesh (Asterisque 396, 2017) , we formulate and prove natural relative branching laws for the restriction of such discrete series representations to smaller groups of the same type and corresponding unitary symmetric spaces. Using period integrals, some results on the Laplacian of spheres, Schlichtkrull et al. (Sao Paulo J Math Sci 12(2), 295–358, 2018) and results by Kobayashi (Adv Math 388, 38, 2021), we prove an analogue of these conjectures for rank-one unitary symmetric spaces.