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On the local Bump–Friedberg L-function

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Published/Copyright: September 26, 2013
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Journal für die reine und angewandte Mathematik (Crelles Journal)
From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2015 Issue 709

Abstract

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 factor L(π,𝐵𝐹,s1,s2) 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),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 [`Derivatives and L-functions for 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].

Received: 2013-2-17
Revised: 2013-7-29
Published Online: 2013-9-26
Published in Print: 2015-12-1

© 2015 by De Gruyter

Articles in the same Issue

  1. Frontmatter
  2. The limit of the Yang–Mills flow on semi-stable bundles
  3. Tangent cones to positive-(1,1) De Rham currents
  4. A characterization of vertex operator algebras V+ℤα: I
  5. Degenerate neckpinches in Ricci flow
  6. On the local Bump–Friedberg L-function
  7. A variational characterization of J-holomorphic curves
  8. Mahler measure and elliptic curve L-functions at s = 3
  9. Uniform bounds for bounded geodesic image theorems
  10. Linear stability of Perelman's ν-entropy on symmetric spaces of compact type
https://doi.org/10.1515/crelle-2013-0083

Articles in the same Issue

  1. Frontmatter
  2. The limit of the Yang–Mills flow on semi-stable bundles
  3. Tangent cones to positive-(1,1) De Rham currents
  4. A characterization of vertex operator algebras V+ℤα: I
  5. Degenerate neckpinches in Ricci flow
  6. On the local Bump–Friedberg L-function
  7. A variational characterization of J-holomorphic curves
  8. Mahler measure and elliptic curve L-functions at s = 3
  9. Uniform bounds for bounded geodesic image theorems
  10. Linear stability of Perelman's ν-entropy on symmetric spaces of compact type
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