## 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 *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_{C} 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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arXiv |
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© Toshiyuki Kobayashi