A converse theorem for quasi-split even special orthogonal groups over finite fields

TL;DR

The paper proves a local converse theorem for quasi-split non-split even special orthogonal groups $SO_{2l}(F)$ over a finite field by introducing twisted gamma-factors $\gamma(\pi\times\tau,\psi)$ with $\tau$ ranging over $GL_n(F)$. The two central technical challenges are the outer automorphism $c$ and the non-split torus, addressed by pairing a representation with its conjugate $\pi^c$ and by refining partial Bessel-function analysis to handle the torus, respectively. Central to the approach are zeta integrals $\Psi(W,f)$, their transformation under an intertwining operator $M$, and a partition of the Bessel support into Bruhat-cell related pieces, together with a hierarchy of twists by $GL_n$ for $n\le l$ (namely $n\le l-2$, $l-1$, and $l$). The main result shows that equality of all twisted gamma-factors for $n\le l$ implies $\pi\cong\pi'$ or $\pi\cong\pi'^c$, aligning with Langlands functoriality and clarifying the role of the outer automorphism in the finite-field setting.

Abstract

We prove a converse theorem for the case of quasi-split non-split even special orthogonal groups over finite fields. There are two main difficulties which arise from the outer automorphism and non-split part of the torus. The outer automorphism is handled similarly to the split case, while new ideas are developed to overcome the non-split part of the torus.

Theorems & Definitions (51)

• Conjecture 1.1
• Theorem 1.2: The Converse Theorem for quasi-split $\mathrm{SO}_{2l}$, Theorem \ref{['converse thm']}
• Theorem 1.3: Theorem \ref{['Bessels Equal']}
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Proposition 3.3
• proof
• Proposition 4.1