On the local Bump–Friedberg L-function

Abstract

Abstract Let F be a p-adic field. If π be an irreducible representation of GL ( n , F ) ${{\rm GL}(n,F)}$ , Bump and Friedberg associated to π an Euler factor L ( π , 𝐵𝐹 , s 1 , s 2 ) ${L(\pi ,\mathit {BF},s_1,s_2)}$ in [Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday. Part II, Weizmann, Jerusalem (1990), 47–65], that should be equal to L ( φ ( π ) , s 1 ) L ( φ ( π ) , Λ 2 , s 2 ) ${L(\phi (\pi ),s_1)L(\phi (\pi ),\Lambda ^2,s_2)}$ , where φ ( π ) ${\phi (\pi )}$ is the Langlands' parameter of π. The main result of this paper is to show that this equality is true when ( s 1 , s 2 ) = ( s + 1 / 2 , 2 s ) $(s_1,s_2)=(s+1/2,2s)$ , for s in ℂ. To prove this, we classify in terms of distinguished discrete series, generic representations of GL ( n , F ) ${{\rm GL}(n,\hspace*{-0.56905pt}F)}$ which are χ α ${\chi _\alpha }$ -distinguished by the Levi subgroup GL ( [ ( n + 1 ) / 2 ] , F ) × GL ( [ n / 2 ] , F ) ${\rm GL}([(n+1)/2], \hspace*{-0.56905pt}F) \times {\rm GL}([n/2], \hspace*{-0.56905pt}F)$ , for χ α ( g 1 , g 2 ) = α ( det ( g 1 ) / det ( g 2 ) ) ${\chi _\alpha (g_1,g_2)=\alpha (\det (g_1)/{\det }(g_2))}$ , where α is a character of F* of real part between - 1 / 2 ${-1/2}$ and 1/2. We then adapt the technique of [`Derivatives and L-functions for GL ( n ) ${\rm GL}(n)$ ', preprint 2011] 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 [Math. Res. Lett. 19 (2012), no. 4, 785–804].