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Symmetry breaking operators for strongly spherical reductive pairs and the Gross-Prasad conjecture for complex orthogonal groups. (arXiv:1705.06109v1 [math.RT])
来源于:arXiv
A real reductive pair $(G,H)$ is called strongly spherical if the homogeneous
space $(G\times H)/{\rm diag}(H)$ is real spherical. This geometric condition
is equivalent to the representation theoretic property that ${\rm
dim\,Hom}_H(\pi|_H,\tau)<\infty$ for all smooth admissible representations
$\pi$ of $G$ and $\tau$ of $H$. In this paper we explicitly construct for all
strongly spherical pairs $(G,H)$ intertwining operators in ${\rm
Hom}_H(\pi|_H,\tau)$ for $\pi$ and $\tau$ spherical principal series
representations of $G$ and $H$. These so-called \textit{symmetry breaking
operators} depend holomorphically on the induction parameters and we further
show that they generically span the space ${\rm Hom}_H(\pi|_H,\tau)$. In the
special case of multiplicity one pairs we extend our construction to
vector-valued principal series representations and obtain generic formulas for
the multiplicities between arbitrary principal series.
As an application, we prove the Gross-Prasad conjecture f 查看全文>>