A local converse theorem for quasi-split even special orthogonal groups

TL;DR

This paper provides a direct proof of the local converse theorem for quasi-split non-split special orthogonal groups over local non-Archimedean fields, utilizing Howe vectors and partial Bessel functions.

Contribution

It introduces a new proof technique for the local converse theorem specific to quasi-split non-split SO groups, expanding the theoretical understanding.

Findings

Proof of the local converse theorem for quasi-split non-split SO groups

Application of Howe vectors and partial Bessel functions in the proof

Enhanced understanding of representation theory for these groups

Abstract

We give a direct proof of the local converse theorem for quasi-split non-split $\mathrm{SO}_{2l}$ over a local non-Archimedean field of characteristic $p\neq 2$, applying the theory of Howe vectors and partial Bessel functions.