The local converse theorem for SO(2n + 1) and applications

Abstract

In this paper we characterize irreducible generic representations of SO 2n+1 (k) (where k is a p-adic field) by means of twisted local gamma factors (the Local Converse Theorem). As applications, we prove that two irreducible generic cuspidal automorphic representations of SO 2n+1 (A) (where A is the ring of adeles of a number field) are equivalent if their local components are equivalent at almost all local places (the Rigidity Theorem); and prove the Local Langlands Reciprocity Conjecture for generic supercuspidal representations of SO 2n+1 (k).

On the nonvanishing of the central value of the Rankin-Selberg 𝐿-functions

“…From now on we simply do not distinguish the Langlands functorial lift from the weak functorial lift. We remark that by the local converse theorem (which has been established by D. Jiang and D. Soudry for SO 2r+1 ([JS03] and [JS04]) and which is their work in progress for other classical groups), one can prove that the representation σ given in the theorem is uniquely determined by π. From the above theorem one deduces…”

A conjecture on Whittaker–Fourier coefficients of cusp forms

“…In the case of SO(2n+1) it was proved that σ is irreducible (see [GRS01], which is based on [JS03]). It follows from the results of [JS07b] that σ is irreducible in the case G = Sp n as well.…”

Langlands Functoriality Conjecture

“…As an application of Theorem F, I repeat again that Jiang and Soudry [38], [39] proved the Local Langlands Reciprocity Law for SO(2n + 1). More precisely, there exists a unique bijective correspondence between the set of conjugacy classes of all 2n-dimensional, admissible, completely reducible, multiplicity-free, symplectic complex representations of the Weil group W k and the set of all equivalence classes of irreducible generic supercuspidal representations of SO(2n + 1, k), which preserves the relevant local factors.…”

“…In both cases, the correspondence was established at the level of a correspondence between irreducible Galois representations and supercuspidal representations. (e) Let k be a a non-archimedean local field of characteristic 0 and let G = SO(2n + 1) the split special orthogonal group over k. In this case, Jiang and Soudry [38], [39] gave a parametrization of generic supercuspidal representations of SO(2n + 1) in terms of admissible homomorphisms of W k . More precisely, there is a unique bijection of the set of conjugacy classes of all admissible, completely reducible, multiplicity-free, symplectic complex representations φ : W k −→ L SO(2n+1) = Sp(2n, C) onto the set of all equivalence classes of irreducible generic supercuspidal representations of SO(2n + 1, k).…”

“…If H = SO(2n + 1), for generic cuspidal representations, Jiang and Soudry [38] proved that the Langlands functorial lift from SO(2n + 1) to GL(2n) is injective up to isomorphism. Using the functorial lifting from SO(2n + 1) to GL(2n), Khare, Larsen and Savin [41] proved that for any prime and any even positive integer n, there are infinitely many exponents k for which the finite simple group P Sp n (F k ) appears as a Galois group over Q.…”