[Paper Review] Stability of Rankin-Selberg gamma factors for $ extrm{Sp}(2n)$, $\widetilde{ extrm{Sp}}(2n)$ and $ extrm{U}(n,n)$

Qing Zhang|arXiv (Cornell University)|Nov 11, 2015

Advanced Algebra and Geometry24 references3 citations

TL;DR

This paper establishes the stability of Rankin-Selberg gamma factors for the symplectic group $\mathrm{Sp}(2n)$, its metaplectic cover $\widetilde{\mathrm{Sp}}(2n)$, and the unitary group $\mathrm{U}(n,n)$ over a $p$-adic field $F$ when the residue characteristic is odd. Using Howe vectors and partial Bessel functions, the authors prove that the gamma factors remain invariant under twisting by highly ramified characters of $F^\times$, extending stability results beyond the Langlands-Shahidi method.

ABSTRACT

Let $F$ be a $p$-adic field and $E/F$ be a quadratic extension. In this paper, we prove the stability of Rankin-Selberg gamma factors for $ extrm{Sp}_{2n}(F)$, $\widetilde { extrm{Sp}}_{2n}(F)$ and $ extrm{U}_{E/F}(n,n)$ when the characteristic of the residue field of $F$ is not $2$.

Motivation & Objective

• To establish the stability of Rankin-Selberg $\gamma$-factors for $\mathrm{Sp}(2n)$, $\widetilde{\mathrm{Sp}}(2n)$, and $\mathrm{U}(n,n)$ over $p$-adic fields.
• To extend stability results to groups not fully covered by the Langlands-Shahidi method, particularly in the context of integral representations.
• To develop a proof framework rooted in integral representations rather than Langlands-Shahidi theory, preserving intrinsic value for L-functions without such constructions.
• To generalize the stability of partial Bessel functions associated with Howe vectors, potentially enabling future local converse theorems for $\mathrm{Sp}(2n)$ and $\mathrm{U}(n,n)$.

Proposed method

• Utilizes Howe vectors constructed via averaging over unipotent subgroups $U_m$ with respect to a generic character $\psi_U$, ensuring invariance under higher congruence subgroups.
• Analyzes partial Bessel functions associated with these Howe vectors, proving their stability under twisting by highly ramified characters.
• Applies the theory of intertwining operators and zeta integrals to relate the $\gamma$-factors to the behavior of Whittaker functions and Schwartz functions.
• Employs gauge estimates and volume normalization to ensure absolute convergence of zeta integrals for $\mathrm{Re}(s)$ sufficiently large.
• Leverages the uniqueness of Fourier-Jacobi models in the unitary group case to establish the existence of $\gamma$-factors as rational functions in $q_F^{-s}$.
• Adapts techniques from Baruch and previous work of the author to extend stability results from $\mathrm{Sp}(2n)$ to $\widetilde{\mathrm{Sp}}(2n)$ and $\mathrm{U}(n,n)$.

Experimental results

Research questions

• RQ1 Does the Rankin-Selberg $\gamma$-factor remain stable under twisting by highly ramified characters of $F^\times$ for $\mathrm{Sp}(2n)$, $\widetilde{\mathrm{Sp}}(2n)$, and $\mathrm{U}(n,n)$ over $p$-adic fields?
• RQ2 Can the stability of $\gamma$-factors be proven directly within the integral representation framework, independent of the Langlands-Shahidi method?
• RQ3 What is the behavior of partial Bessel functions associated with Howe vectors under twisting by highly ramified characters?
• RQ4 Under what conditions on the residue characteristic and extension $E/F$ does the stability result hold for $\mathrm{U}(n,n)$?
• RQ5 Can the stability of partial Bessel functions lead to a local converse theorem for $\mathrm{Sp}(2n)$ and $\mathrm{U}(n,n)$?

Key findings

• The $\gamma$-factors for $\mathrm{Sp}(2n)$, $\widetilde{\mathrm{Sp}}(2n)$, and $\mathrm{U}(n,n)$ are stable under twisting by highly ramified characters of $F^\times$ when the residue characteristic of $F$ is odd.
• The stability of partial Bessel functions associated with Howe vectors is established as a key technical result, with Theorem 3.11 providing the core analytic tool.
• For $\mathrm{U}(n,n)$, the result holds when $E/F$ is unramified or when $E/F$ is ramified but the residue characteristic is not 2.
• The proof is self-contained within the Rankin-Selberg integral framework, avoiding reliance on Langlands-Shahidi $\gamma$-factors, even though agreement with them is known in some cases.
• The method yields a stability result for $\mathrm{U}(n,n)$ even though the agreement between Rankin-Selberg and Langlands-Shahidi $\gamma$-factors is not established in the cited literature.
• The stability of $\gamma$-factors is shown to hold for all $m \geq C$ where $C$ is an integer such that the vectors are fixed by $K_C$, ensuring uniform behavior in the limit.