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Group Gradings on Lie Algebras and Applications to Geometry: II Read chapter Read first chapter

2014 | OriginalPaper | Chapter

Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators

Author : Toshiyuki Kobayashi

Published in: Developments and Retrospectives in Lie Theory

Publisher: Springer International Publishing

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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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previous chapter Cycle Connectivity and Automorphism Groups of Flag Domains
next chapter Harmonic Spinors on Reductive Homogeneous Spaces
Footnotes
1
The author was partially supported by Grant-in-Aid for Scientific Research (B)(22340026) and (A)(25247006).
 
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Metadata
Title
Shintani Functions, Real Spherical Manifolds, and Symmetry Breaking Operators
Author
Toshiyuki Kobayashi
Copyright Year
2014
Book
Developments and Retrospectives in Lie Theory

Print ISBN: 978-3-319-09933-0

Electronic ISBN: 978-3-319-09934-7

Copyright Year: 2014

DOI
https://doi.org/10.1007/978-3-319-09934-7_5

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