[Paper Review] On the local Bump-Friedberg L-function

Nadir Matringe|arXiv (Cornell University)|Jan 2, 2013

Advanced Algebra and Geometry4 citations

TL;DR

This paper establishes that the local Bump-Friedberg L-function for an irreducible representation $π$ of $GL(n,F)$, where $F$ is a $p$-adic field, equals $L(\phi(\pi),s+1/2)L(\phi(\pi),\Lambda^2,2s)$, with $\phi(\pi)$ the Langlands parameter of $\pi$. The proof relies on classifying $\chi_\alpha$-distinguished generic representations and reducing the identity to discrete series via Whittaker and Shalika period relations, ultimately confirming the conjectured factorization for all irreducible representations.

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 fator $L(π,BF,s_1,s_2)$ in \cite{BF}, that should be equal to $L(ϕ(π),s_1)L(ϕ(π),Λ^2,s_2)$, where $ϕ(π)$ is the Langlands' parameter of $π$. The main result of this paper is to show that this equality is true when $(s_1,s_2)=(s+1/2,2s)$, for $s$ in $\C$. To prove this, we classify in terms of distinguished discrete series, generic representations of $GL(n,F)$ which are $χ_α$-distinguished by the Levi subgroup $GL([(n+1)/2],F) imes GL([n/2],F)$, for $χ_α(g_1,g_2)=α(det(g_1)/det(g_2))$, where $α$ is a character of $F^*$ of real part between -1/2 and 1/2. We then adapt the technique of \cite{CP} 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 \cite{KR}.

Motivation & Objective

• To verify the conjectured equality between the local Bump-Friedberg $L$-function and the product of standard $L$-functions $L(\phi(\pi),s+1/2)L(\phi(\pi),\Lambda^2,2s)$ for irreducible representations of $GL(n,F)$.
• To classify generic representations of $GL(n,F)$ that are $\chi_\alpha$-distinguished by the Levi subgroup $GL([(n+1)/2],F)\times GL([n/2],F)$, where $\chi_\alpha(g_1,g_2) = \alpha(\det g_1 / \det g_2)$ and $\alpha$ has real part in $[-1/2, 1/2]$.
• To extend the functional equation and rationality results for Rankin-Selberg integrals associated with Whittaker-type representations.
• To establish the equality for discrete series representations using the relation between linear and Shalika periods, and then extend it to all irreducible representations via induction and Langlands classification.

Proposed method

• Classify $\chi_\alpha$-distinguished generic representations of $GL(n,F)$ in terms of $\chi_\alpha$-distinguished discrete series, using results from [27] and properties of Bernstein-Zelevinsky derivatives.
• Adapt techniques from [10] to reduce the proof of the $L$-function equality to the case of discrete series representations.
• Use the relation between linear periods and Shalika periods for unitary discrete series to equate $L^{\text{lin}}(\Delta,s)$ with $L(\phi(\Delta),s+1/2)L(\Lambda^2(\phi(\Delta)),2s)$.
• Prove rationality of Rankin-Selberg integrals twisted by unramified characters of the Levi subgroup, establishing the functional equation and pole structure.
• Apply induction on the discrete series parameter $k$ in $\Delta = \text{St}_k(\rho)$, using the inductivity of $L^{\text{lin}}(\pi,\chi_\alpha,s)$ and the product formulas for $L(\phi(\Delta),\wedge^2,s)$.
• Leverage the fact that exceptional poles of $L^{\text{lin}}(\pi,\chi_\alpha,s)$ occur precisely when $\pi$ is $\chi_\alpha^{-1}$-distinguished, linking poles to the classification of distinguished representations.

Experimental results

Research questions

• RQ1 Does the local Bump-Friedberg $L$-function $L(\pi,BF,s_1,s_2)$ satisfy $L(\pi,BF,s+1/2,2s) = L(\phi(\pi),s+1/2)L(\phi(\pi),\Lambda^2,2s)$ for all irreducible $\pi$ of $GL(n,F)$?
• RQ2 Which generic representations of $GL(n,F)$ are $\chi_\alpha$-distinguished for $\chi_\alpha(g_1,g_2) = \alpha(\det g_1 / \det g_2)$ with $\text{Re}(\alpha) \in [-1/2, 1/2]$?
• RQ3 How do the linear periods $L^{\text{lin}}(\pi,s)$ relate to the standard $L$-functions $L(\phi(\pi),s+1/2)$ and $L(\phi(\pi),\Lambda^2,2s)$ for discrete series representations?
• RQ4 Can the equality $L^{\text{lin}}(\pi,s) = L(\phi(\pi),s+1/2)L(\phi(\pi),\Lambda^2,2s)$ be extended from discrete series to all irreducible representations via induction and Langlands classification?

Key findings

• The equality $L^{\text{lin}}(\Delta,s) = L(\phi(\Delta),s+1/2)L(\phi(\Delta),\wedge^2,2s)$ holds for all discrete series representations $\Delta$ of $GL(n,F)$, proven via the relation between linear and Shalika periods and the main result of [20].
• For $\chi_\alpha$-distinguished generic representations with $\text{Re}(\alpha) \in [-1/2, 1/2]$, the classification shows they are $\chi_\alpha$-distinguished if and only if their Langlands parameter preserves a certain structure, generalizing the symplectic condition in the trivial character case.
• The factor $L^{\text{lin}}(\pi,\chi_\alpha,s)$ satisfies an inductivity relation for representations of Langlands type when $\text{Re}(\alpha) \in [-1/2, 0]$, enabling reduction to discrete series.
• The rationality of the Rankin-Selberg integrals $\Psi(W,\phi,s)$ is established when twisted by unramified characters of the Levi subgroup, ensuring meromorphic continuation and functional equations.
• The exceptional pole of $L^{\text{lin}}(\pi,\chi_\alpha,s)$ at $s=0$ occurs if and only if $\pi$ is $\chi_\alpha^{-1}$-distinguished, linking pole structure to the classification of distinguished representations.
• The main result is extended to all irreducible representations $\tau$ of $GL(n,F)$, yielding $L^{\text{lin}}(\tau,s) = L(\phi(\tau),s+1/2)L(\phi(\tau),\Lambda^2,2s)$, confirming the Bump-Friedberg conjecture for the specified parameters.