On the local converse Theorem for split $\mathrm{SO}_{2l}$

TL;DR

This paper proves the local converse theorem for split even special orthogonal groups over non-Archimedean fields, addressing the challenge posed by outer automorphisms, and applies the result to a rigidity theorem for certain representations.

Contribution

It establishes the local converse theorem for split even special orthogonal groups, filling a key gap in the theory of split classical groups.

Findings

Proves the local converse theorem for split SO(2l) groups.

Introduces new methods involving partial Bessel functions.

Derives a weak rigidity theorem for generic cuspidal representations.

Abstract

In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic zero. This is the only case left on local converse theorems of split classical groups and the difficulty is the existence of the outer automorphism. We apply new ideas of considering the summation of partial Bessel functions and overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups.