On a refined local converse theorem for SO(4)

TL;DR

This paper refines the local converse theorem for SO(4) over p-adic fields, showing that a generic supercuspidal representation is uniquely determined by specific twisted gamma factors, removing the need for an outer automorphism consideration.

Contribution

It provides a solution for SO(4) that determines generic supercuspidal representations solely by certain twisted gamma factors, improving previous results that required considering outer automorphisms.

Findings

Unique determination of supercuspidal representations by gamma factors.

Elimination of the outer automorphism ambiguity for SO(4).

Enhanced understanding of local gamma factors in representation theory.

Abstract

Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local converse theorem for $\mathrm{SO}_{2n}(F)$ over a $p$-adic field $F$, which says that, up to an outer automorphism of $\mathrm{SO}_{2n}(F)$, an irreducible generic representation of $\mathrm{SO}_{2n}(F)$ is uniquely determined by its twisted gamma factors by generic representations of $\mathrm{GL}_k(F)$ for $k=1,\dots,n$. It is desirable to remove the ``up to an outer automorphism" part in the above theorem using more twisted gamma factors, but this seems a hard problem. In this paper, we provide a solution to this problem for the group $\mathrm{SO}_4(F)$, namely, we show that a generic supercuspidal representation $\pi$ of $\mathrm{SO}_4(F)$ is uniquely determined by its $\mathrm{GL}_1$, $\mathrm{GL}_2$ twisted local gamma factors and a twisted exterior square local gamma factor of $\pi$.