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On the local Langlands correspondence for non-tempered representations. (English) Zbl 1408.22017
Let \(G\) be a reductive \(p\)-adic group. The local Langlands correspondence (LLC) conjectures the existence of a map \[ \mathrm{Irr}(G) \to \Phi(G) \] satisfying several naturality properties. Here, \(\mathrm{Irr}(G)\) is the set of equivalence classes of irreducible smooth representations of \(G\) and \(\Phi(G)\) is the set of conjugacy classes of Langlands parameters for \(G\). Let \(\mathrm{Irr}^t(G)\) be the subset of \(\mathrm{Irr}(G)\) consisting of tempered representations. According to Langlands’ conjectures, \(\mathrm{Irr}^t(G)\) corresponds to the set \(\Phi_{\mathrm{bdd}}(G)\) of bounded parameters. The aim of this paper is to show, assuming the LLC for \(\mathrm{Irr}^t(G)\), how to obtain the LLC for \(\mathrm{Irr}(G)\).
If \(\phi \in \Phi(G)\) is an \(L\)-parameter, the corresponding \(L\)-packet \(\Pi_\phi(G) \subset \mathrm{Irr}(G)\) should be finite and its elements should be parametrized by irreducible representations \(\rho\) of a finite group \(\mathcal{S}_\phi\). Let \(\Phi^e(G)\) denote the set of enhanced Langlands parameters \((\phi, \rho)\). Then the LLC should become an injection \[ \mathrm{Irr}(G) \to \Phi^e(G). \] The main result in the paper is the following theorem.
Theorem. Suppose that a tempered local Langlands correspondence is given as an injective map \(\mathrm{Irr}^t(G) \to \Phi_{\mathrm{bdd}}^e(G)\), which is compatible with twisting by unramified characters whenever this is well defined. Then the map extends canonically to a local Langlands correspondence \(\mathrm{Irr}(G) \to \Phi^e(G).\)
For the proof, the authors introduce new \(R\)-groups and study their properties. These \(R\)-groups generalize Knapp-Stein \(R\)-groups. They are associated to non-tempered essentially square integrable representations of Levi subgroups of \(G\).

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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