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 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)).
© Toshiyuki Kobayashi