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Generic representations and local Langlands reciprocity law for \(p\)-adic \(\text{SO}_{2n+1}\). (English) Zbl 1062.11077

Hida, Haruzo (ed.) et al., Contributions to automorphic forms, geometry, and number theory. Papers from the conference in honor of Joseph Shalika on the occasion of his 60th birthday, Johns Hopkins University, Baltimore, MD, USA, May 14–17, 2002. Baltimore, MD: Johns Hopkins University Press (ISBN 0-8018-7860-8/hbk). 457-519 (2004).
Let \(G\) denote the split \(\text{SO}(2n+1)\) over a non-Archimedean local field \(F\) of characteristic 0. The L-group of \(G\) is \(\text{Sp}(2n,{\mathbb C})\), so the set of Langlands parameters for \(G\) is a subset of the set of Langlands parameters for \(\text{GL}(2n)\) and it is known that the latter set is in one-one correspondence with the set of all (equivalence classes of) irreducible smooth representations of \(\text{GL}(2n,F)\). Thus a partition of the set of all irreducible smooth representations of \(G(F)\) into L-packets would imply a map from L-packets for \(G\) to representations of \(\text{GL}(2n,F)\), the functorial lifting of representations.
In their earlier paper [Ann. Math. (2) 157, 743–806 (2003; Zbl 1049.11055)] the authors determined explicitly the lift of generic irreducible supercuspidal representations of \(G(F)\). In the present paper they extend this in several steps to all generic irreducible representations of \(G(F)\) (Theorem 5.1). In particular, the Langlands parameters of these representations are determined.
Another theorem assigns to any Langlands parameter \(\phi\) a representation \(y(\phi)\) of \(G(F)\), preserving twisted L- and \(\varepsilon\)-factors (Theorem 6.1). A criterion for genericity of \(y(\phi)\) is given.
Two applications to automorphic representations are given.
For the entire collection see [Zbl 1051.11005].

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
11F70 Representation-theoretic methods; automorphic representations over local and global fields
22E50 Representations of Lie and linear algebraic groups over local fields

Citations:

Zbl 1049.11055
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