On the local converse theorem for split 𝑆𝑂_{2𝑙}

In this paper, we prove the local converse theorem for split even special orthogonal groups over a non-Archimedean local field of characteristic <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="p not-equals 2"> <mml:semantics> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">p\neq 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. 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 a new idea by considering a certain sum of partial Bessel functions to overcome this difficulty. As a direct application, we obtain a weak rigidity theorem for irreducible generic cuspidal representations of split even special orthogonal groups.