On twisted exterior and symmetric square \gamma -factors

Abstract

We establish the existence and uniqueness of twisted exterior and symmetric square γ-factors in positive characteristic by studying the Siegel Levi case of generalized spinor groups. The corresponding theory in characteristic zero is due to Shahidi. In addition, in characteristic p we prove that these twisted local factors are compatible with the local Langlands correspondence. As a consequence, still in characteristic p, we obtain a proof of the stability property of γ-factors under twists by highly ramified characters. Next we use the results on the compatibility of the Langlands-Shahidi local coefficients with the Deligne-Kazhdan theory over close local fields to show that the twisted symmetric and exterior square γ-factors, L-functions and ε-factors are preserved over close local fields. Furthermore, we obtain a formula for Plancherel measures in terms of local factors and we also show that they also preserved over close local fields.

Rankin–selberg Integrals for Local Symmetric Square Factors on Gl(2)

“…The analogous result has been settled for over a non‐Archimedean field F with characteristic zero [8] and with positive characteristic [9] in the formulation of Langlands–Shahidi local coefficients.…”

“…Proof The induced representation can be viewed as a subrepresentation of a genuine principal series representation , where is an extension to of the genuine character of defined by for . Shahidi [9, 10, 34] defines the Plancherel measure associated with η by It is a priori a rational function in . As described in [10, (4.9),(4.11),(9.22)], the formula we seek for …”

“…As elucidated in the Langlands–Shahidi method [8, 9, 34], the unit appearing in Proposition 4.7‐(ii) will be presumably 1. This is so‐called the multiplicativity of γ‐factors.…”

“…The Rankin–Selberg method plays a profound role in analyzing Langlands automorphic L ‐functions in company with the Langlands–Shahidi method [8–10, 34] and the doubling method [13, 43]. The main subject of the Rankin–Selberg convolution is to determine the integral representation which admits a factorization into an Euler product.…”

On the Local Langlands Correspondence for Split Classical Groups Over Local Function Fields

“…(VIII) (Symmetric and exterior square local factors) Using (V), it can be shown that the symmetric and exterior square L-and γ -factors are the same for sufficiently close local fields. More precisely, let (π, V ) be a representation of GL n (F) with depth(π ) < m, and let ψ be a nontrivial additive character of F. It was shown in [30] that there exists l = l(m, n) m + 1 such that, for any field F that is l-close to F, the representation π of GL n (F ) with π ∼ l π , and character ψ of F with ψ ∼ l ψ, we have Then each nondegenerate character of U is T -conjugate to χ a for some a ∈ F × . For H n = SO 2n+1 , we in fact have that each nondegenerate character is conjugate to χ 1 .…”