Uniqueness of Rankin–Selberg products

Abstract

Abstract. In the present paper, we show the equality of the γ-factors defined by Jacquet, Piatetski-Shapiro and Shalika with those obtained via the Langlands-Shahidi method. Contrary to the local proof given by Shahidi, our proof uses a refined version of the local-global principle for GLn in positive characteristic, which has independent interest. The comparison of γ-factors is made via a uniqueness result for Rankin-Selberg γ-factors over a non-Archimedean local field of positive characteristic.

The local Langlands correspondence for GSp 4 over local function fields

“…By [45] (Also cf. [23] for an alternate proof), we know that the γ-factor arising out of the Langlands-Shahidi method (which is simply the associated local coefficient) agrees with the Rankin-Selberg γ-factor. Hence we only need to check the l chosen in the statement above is large enough for the hypothesis of Theorem 5.5 to be satisfied for all representations of σ of depth ≤ m. View GL n (F ) × GL t (F ) as a maximal Levi subgroup of GL n+t (F ).…”

On twisted exterior and symmetric square \gamma -factors

“…For uniqueness, let γ be a rule on L(p) satisfying the hypothesis of the theorem. Given (F, π, η, ψ) ∈ L(p), Properties (v) and (vi) give that γ(s, π, r 0 ⊠ η, ψ) depends only on the supercuspidal content of π (see Remark 2.3 of [20]). Fix a supercuspidal quadruple (F, π, η, ψ) of L(p).…”

“…Fix a supercuspidal quadruple (F, π, η, ψ) of L(p). Theorem 3.1 of [20] gives a global field k, a cuspidal automorphic representation ξ = ⊗ ξ v of GL n (A k ) such that: ξ 0 ≃ π and ξ v , v = 0 is a subquotient of a principal series representation. Applying the Grundwald-Wang theorem of class field theory gives us a Grössencharakter ν = ⊗ν v such that ν 0 ≃ η.…”

“…Furthermore, we have that µ(s, τ i,j ,w i,j ) = µ(s, π i × (π j · η),w ′ 0 ). Plancherel measures for general linear groups and the Langlands-Shahidi local coefficient are explored in [32], the connection to Rankin-Selberg γ-factors in characteristic p can be made through the results of [20].…”

“…In Theorem 3.1.1 below, which addresses the characteristic p case, we extend the results of [18] on exterior and symmetric square γ-factors. The existence part is possible via the results of [29] over function fields and the uniqueness argument is due to Henniart-Lomelí, thanks to the local-global technique for GL n found in [18,20].…”