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Branching laws for classical groups: the non-tempered case. (English) Zbl 1470.11126

Consider representations of a classical group \(G\) over a local field \(k\) arising as local components of automorphic representations in the automorphic dual and with \(A\)-paramaters (Arthur parameters) of the form \[\phi_A:WD(k)\times SL_2({\mathbb C})\rightarrow\mbox{}^L G\] where the restriction of \(\phi_A\) to the Weil-Deligne group \(WD(k)\) of \(k\) is an admissible homomorphism with bounded image and the restriction to \(SL_2\) is algebraic. Recall that the abelianization of the Weil group is isomorphic to \(k^*\). Now, composing \(\phi_A\) with the map \[w\mapsto (w,\text{diag}(\mid w\mid^{1/2},\mid w\mid^{-1/2}))\] where \(\mid\;\mid\) denotes the canonical absolute value on \(k^*\), one gets an injection \(\phi_A\mapsto\phi\) from the set of \(A\)-parameters to the set of \(L\)-parameters. Such \(L\)-parameters are known as \(L\)-parameters of Arthur type. When the \(L\)-parameter \(\phi\) is tempered, the associated \(L\)-packet is generic.
Earlier work by the authors suggested several precise conjectures, in both the local and the global cases, for restrictions of representations lying in local \(L\)-packets whose \(L\)-parameters are generic. In the paper under review, the authors discuss a generalization of these conjectures to certain non-generic \(L\)-packets.
Reviewer: Salah Mehdi (Metz)

MSC:

11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings

Citations:

Zbl 1280.22019
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References:

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