Shintani functions, real spherical manifolds, and symmetry breaking operators

Abstract

For a pair of reductive groups $G\supset G^{\prime }$, 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 $(G,G^{\prime })$ having finite-dimensional Shintani spaces. A geometric criterion for uniform boundedness of $dimSh(\lambda ,\nu )$ 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 $(G,G^{\prime })=(O(n+1,1),O(n,1))$.

to appear in Progress in Mathematics, Birkhauser

References in corpus (3)

• Symmetry breaking for representations of rank one orthogonal groups II
• Multiplicity one theorem for (GL(n+1,R),GL(n,R))
• Classification of finite-multiplicity symmetric pairs

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