A Converse Theorem for Split $\mathrm{SO}_{2l}$ over Finite Fields

TL;DR

This paper establishes a converse theorem for split even special orthogonal groups over finite fields, addressing a key open case in the theory of split classical groups by overcoming challenges posed by outer automorphisms.

Contribution

It introduces new methods to prove a converse theorem for split SO_{2l} over finite fields, handling the complexity of outer automorphisms.

Findings

Proves a converse theorem for split SO_{2l} over finite fields.

Develops new techniques to handle outer automorphisms.

Completes the classification of split classical groups with respect to converse theorems.

Abstract

We prove a converse theorem for split even special orthogonal groups over finite fields. This is the only case left on converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. In this paper, we develop new ideas and overcome this difficulty.