2013/01/02 by Nadir Matringe, Matringe, Nadir · 1 citation
Mathematics · #11F66 #11F70 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1301.0350
openalex publication_date 2013/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let F be a p-adic field. If π be an irreducible representation of GL(n,F), Bump and Friedberg associated to π an Euler fator L(π,BF,s1,s2) in \citeBF, that should be equal to L(ϕ(π),s1)L(ϕ(π),Λ2,s2), where ϕ(π) is the Langlands' parameter of π. The main result of this paper is to show that this equality is true when (s1,s2)=(s+1/2,2s), for s in \C. To prove this, we classify in terms of distinguished discrete series, generic representations of GL(n,F) which are χα-distinguished by the Levi subgroup GL([(n+1)/2],F) × GL([n/2],F), for χα(g1,g2)=α(det(g1)/det(g2)), where α is a character of F^* of real part between -1/2 and 1/2. We then adapt the technique of \citeCP to reduce the proof of the equality to the case of discrete series. The equality for discrete series is a consequence of the relation between linear periods and Shalika periods for discrete series, and the main result of \citeKR.