Shintani functions, real spherical manifolds, and symmetry breaking   operators
  For a pair of reductive groups $G \supset G'$, we prove a geometric criterion for the space $Sh(\lambda, \nu)$ 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 $dim Sh(\lambda, \nu)$ 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 $(G, G') = (O(n+1,1), O(n,1))$. 
Toshiyuki Kobayashi
math.RT math.GR math.NT