ON A REFINED LOCAL CONVERSE THEOREM FOR SO(4)

Recently, Hazeltine-Liu, and independently Haan-Kim-Kwon, proved a local converse theorem for SO2n(F)overapadic field F,which says that, up to an outer automorphism of SO2n(F), an irreducible generic representation of SO2n(F) is uniquely determined by its twisted gamma factors by generic representations of GL(k)(F)for k=1,...,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 SO4(F), namely, we show that a generic super cuspidal representation pi of SO4(F) is uniquely determined by its GL(1),GL(2 )twisted local gamma factors and a twisted exterior square local gamma factor of pi.