[Paper Review] On the Local Converse Theorem for p-adic GLn

Hervé Jacquet, Baiying Liu|arXiv (Cornell University)|Jan 14, 2016

Advanced Algebra and Geometry18 references3 citations

TL;DR

This paper proves the local converse theorem for generic representations of $\mathrm{GL}_n(F)$ over a non-archimedean local field $F$, establishing that the equality of local gamma factors $\gamma(s, \pi_1 \times \tau, \psi) = \gamma(s, \pi_2 \times \tau, \psi)$ for all irreducible generic $\tau$ of $\mathrm{GL}_r(F)$ with $1 \leq r \leq \lfloor n/2 \rfloor$ implies $\pi_1 \cong \pi_2$. The proof uses analytic methods based on Fourier inversion and integral identities, without relying on special Whittaker function pairs, and confirms a long-standing conjecture in the theory of automorphic forms and Langlands functoriality.

ABSTRACT

In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.

Motivation & Objective

• To resolve a standard conjecture in the theory of automorphic forms concerning the determination of irreducible generic representations of $\mathrm{GL}_n(F)$ by their local gamma factors.
• To establish that the family of gamma factors $\gamma(s, \pi \times \tau, \psi)$ for $\tau$ of $\mathrm{GL}_r(F)$ with $1 \leq r \leq \lfloor n/2 \rfloor$ uniquely determines $\pi$.
• To provide a new analytic proof of the local converse theorem that avoids the use of special pairs of Whittaker functions, previously used in related works.
• To confirm that the bound $r = \lfloor n/2 \rfloor$ is sharp for the generic dual of $\mathrm{GL}_n(F)$, as supported by prior constructions and counterexamples.

Proposed method

• The proof employs Fourier inversion on unipotent subgroups associated with the Whittaker model of generic representations.
• It uses integral identities involving the Whittaker functions of two generic representations $\pi_1$ and $\pi_2$ under the assumption that their gamma factors agree for all $\tau$ of $\mathrm{GL}_r(F)$ with $r \leq \lfloor n/2 \rfloor$.
• The method applies descending induction on the size of the unipotent group, reducing the equality of integrals over increasingly smaller subgroups.
• By transforming variables between $X$ and $\widetilde{X}$, the proof shows that the Jacobian determinant satisfies $|\widetilde{p}(\widetilde{X})p(X)| = 1$, preserving the measure under change of variables.
• The key step involves applying the Fourier inversion formula on the group $X_k$ to equate integrals over smaller subgroups, ultimately reducing to the equality of integrals over the trivial group.
• The argument relies on the structure of upper-triangular unipotent matrices and the behavior of additive characters under conjugation and multiplication.

Experimental results

Research questions

• RQ1 Does the equality of local gamma factors $\gamma(s, \pi_1 \times \tau, \psi) = \gamma(s, \pi_2 \times \tau, \psi)$ for all $\tau$ of $\mathrm{GL}_r(F)$ with $1 \leq r \leq \lfloor n/2 \rfloor$ imply that $\pi_1 \cong \pi_2$ for irreducible generic representations $\pi_1, \pi_2$ of $\mathrm{GL}_n(F)$?
• RQ2 Is the bound $r = \lfloor n/2 \rfloor$ sharp for the local converse theorem in the generic case, or could a smaller $r$ suffice?
• RQ3 Can the local converse theorem be proven without constructing special pairs of Whittaker functions, as previously done in related works?
• RQ4 Does the global version of the local converse theorem follow from the local result, given the known implications in the literature?
• RQ5 Is the sharpness of $\lfloor n/2 \rfloor$ preserved in the supercuspidal case, particularly for prime $n$?

Key findings

• The paper fully proves Conjecture 1.1: if two irreducible generic representations $\pi_1, \pi_2$ of $\mathrm{GL}_n(F)$ have the same central character and identical gamma factors $\gamma(s, \pi_i \times \tau, \psi)$ for all $\tau$ of $\mathrm{GL}_r(F)$ with $1 \leq r \leq \lfloor n/2 \rfloor$, then $\pi_1 \cong \pi_2$.
• The proof establishes that the bound $r = \lfloor n/2 \rfloor$ is sharp for the generic dual of $\mathrm{GL}_n(F)$, as demonstrated by explicit counterexamples in the literature.
• The method avoids the use of special Whittaker function pairs, offering a new analytic approach based on Fourier inversion and measure preservation under variable transformation.
• The authors show that the equality of integrals over unipotent subgroups, after successive applications of Fourier inversion, implies the isomorphism of the representations.
• The result confirms that the local Langlands correspondence for $\mathrm{GL}_n(F)$ can be uniquely determined by gamma factors up to $r = \lfloor n/2 \rfloor$, reducing the number of required factors.
• The work provides strong evidence for the global converse theorem, as the local result implies the global version under known functoriality constraints.