Abstract
For a pair of reductive groups G ⊃ G′, 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 \(\dim _{\mathbb{C}}\mathrm{Sh}(\lambda,\nu )\) 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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Notes
- 1.
The author was partially supported by Grant-in-Aid for Scientific Research (B)(22340026) and (A)(25247006).
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Acknowledgements
Parts of the results and the idea of the proof were delivered in various occasions including Summer School on Number Theory in Nagano (Japan) in 1995 organized by F. Sato, Distinguished Sackler Lectures organized by J. Bernstein at Tel Aviv University (Israel) in May 2007, the conference in honor of E. B. Vinberg’s 70th birthday in Bielefeld (Germany) in July 2007 organized by H. Abels, V. Chernousov, G. Margulis, D. Poguntke, and K. Tent, “Lie Groups, Lie Algebras and their Representations” in November 2011 in Berkeley (USA) organized by J. Wolf, “Group Actions with Applications in Geometry and Analysis” at Reims University (France) in June, 2013 organized by J. Faraut, J. Hilgert, B. Ørsted, M. Pevzner, and B. Speh, and “Representations of Reductive Groups” at University of Utah (USA) in July, 2013 organized by J. Adams, P. Trapa, and D. Vogan. The author is grateful for their warm hospitality.
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Kobayashi, T. (2014). Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators. In: Mason, G., Penkov, I., Wolf, J. (eds) Developments and Retrospectives in Lie Theory. Developments in Mathematics, vol 37. Springer, Cham. https://doi.org/10.1007/978-3-319-09934-7_5
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