On the local Bump-Friedberg L function II

TL;DR

This paper proves that a specific local L-function for representations of p-adic GL(n) matches the expected Langlands' factor, extending previous work and connecting distinguished representations with Shalika models.

Contribution

It establishes the equality of the local Euler factor with the Langlands' expected factor for a broader class of representations, generalizing previous results.

Findings

The local Euler factor matches the Langlands' predicted factor.

Discrete series representations are characterized by the existence of Shalika models.

Extension of distinguished representation criteria to non-trivial characters.

Abstract

Let $F$ be a $p$-adic field with residue field of cardinality $q$. To each irreducible representation of $GL(n,F)$, we attach a local Euler factor $L^{BF}(q^{-s},q^{-t},\pi)$ via the Rankin-Selberg method, and show that it is equal to the expected factor $L(s+t+1/2,\phi_\pi)L(2s,\Lambda^2\circ \phi_\pi)$ of the Langlands' parameter $\phi_\pi$ of $\pi$. The corresponding local integrals were introduced in [BF], and studied in [M15]. This work is in fact the continuation of [M15]. The result is a consequence of the fact that if $\delta$ is a discrete series representation of $GL(2m,F)$, and $\chi$ is a character of Levi subgoup $L=GL(m,F)\times GL(m,F)$, trivial on $GL(m,F)$ embedded diagonally, then $\delta$ is $(L,\chi)$-distinguished if an only if it admits a Shalika model, a result which was only established for $\chi=1$ before.