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On the decomposition of a representation of $$SO_ n$$ when restricted to $$SO_{n-1}$$. (English) Zbl 0787.22018
Let $$k$$ be a local field, $$\text{char}(k)\neq 2$$, and let $$SO(V)$$ be the special orthogonal group of a quadratic space over $$k$$, i.e. a finite dimensional vector space over $$k$$ with a non-degenerate quadratic form $$q: V\to k$$. Given a vector $$v\in V$$, $$q(v)\neq 0$$, let $$SO(W)$$, $$W=\langle v\rangle^ \perp$$, the subgroup of $$SO(V)$$ which fixes the vector $$v$$. Let $$\pi= \pi_ 1\otimes \pi_ 2$$ be an irreducible representation of the group $$G:= SO(V)(k) \times SO(W)(k)$$, where $$\pi_ 1$$ (resp. $$\pi_ 2$$) is an irreducible representation of $$SO(V)(k)$$ (resp. $$SO(W)(k)$$). The group $$H=SO(W)(k)$$ embeds diagonally as a subgroup of $$G$$. The problem of determining the dimension of $$\operatorname{Hom}_ H(\pi,\mathbb{C})$$ turns out to be a reformulation of the natural problem to study the restriction of irreducible admissible complex representations of $$SO(V)(k)$$ to $$SO(W)(k)$$.
As shown by I. Piatetski-Shapiro and S. Rallis $$\dim\operatorname{Hom}_ H(\pi,\mathbb{C})\leq 1$$, so the question is to identify those $$\pi$$ which admit a non-trivial $$H$$-invariant linear form. The answer is still conjectural but verified in many cases. The formulation is based on Langlands parametrization of irreducible representations of $$G$$, in Vogan’s revised form, and it involves the local root numbers of symplectic representations of the Weil-Deligne group of $$k$$. The background of this work is a question of restriction of irreducible automorphic representations, which is related to central critical values of $$L$$-functions.

##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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