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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 $$G$$.

##### 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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