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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.

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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