T. Kobayashi. Shintani functions, real spherical manifolds, and symmetry breaking operators. In G. Mason, I. Penkov, and Joseph A. Wolf, editors, Developments and Retrospectives in Lie Theory Geometric and Analytic Methods, volume 37 of Developments in Mathematics, pages 127-159, 2014. arXiv: 1401.0117. DOI: 10.1007/978-3-319-09934-7_5..

For a pair of reductive groups GG', we prove a geometric criterion for the space Sh(λ, ν) of Shintani functions to be finite-dimensional in the Archimedean case. This criterion leads us to a complete classification of the symmetric pairs (G,G') having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of dimC Sh(λ, ν) is also obtained. Furthermore, we prove that symmetry breaking operators of the restriction of smooth admissible representations yield Shintani functions of moderate growth, of which the dimension is determined for (G, G') = (O(n+1,1), O(n,1)).

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