Mod \(\ell\) gamma factors and a converse theorem for finite general linear groups

• Published in: arXiv.org 2023-07
• Main Authors: Bakeberg, Jacksyn, Gerbelli-Gauthier, Mathilde, Goodson, Heidi, Iyengar, Ashwin, Moss, Gilbert, Zhang, Robin
• Format: Article
• Language: English
• Subjects: Algebra; Representations; Theorems
• ISSN: 2331-8422

For \(q\) a power of a prime \(p\), we study gamma factors of representations of \(GL_n(\mathbb{F}_q)\) over an algebraically closed field \(k\) of positive characteristic \(\ell \neq p\). We show that the reduction mod \(\ell\) of the gamma factor defined in characteristic zero fails to satisfy the analogue of the local converse theorem of Piatetski-Shapiro. To remedy this, we construct gamma factors valued in arbitrary \(\mathbb{Z}[1/p, \zeta_p]\)-algebras \(A\), where \(\zeta_p\) is a primitive \(p\)-th root of unity, for Whittaker-type representations \(\rho\) and \(\pi\) of \(GL_n(\mathbb{F}_q)\) and \(GL_m(\mathbb{F}_q)\) over \(A\). We let \(P(\pi)\) be the projective envelope of \(\pi\) and let \(R(\pi)\) be its endomorphism ring and define new gamma factors \(\widetilde\gamma(\rho \times \pi) = \gamma((\rho\otimes_kR(\pi)) \times P(\pi))\), which take values in the local Artinian \(k\)-algebra \(R(\pi)\). We prove a converse theorem for cuspidal representations using the new gamma factors. When \(n=2\) and \(m=1\) we construct a different ``new'' gamma factor \(\gamma^{\ell}(\rho,\pi)\), which takes values in \(k\) and satisfies a converse theorem.