Fourier–Mukai transformations on K3 surfaces with ρ=1ρ=1 and Atkin–Lehner involutions

We show that there is a surjection from the Fourier–Mukai transformations on projective K3 surfaces with the Picard number ρ(X)=1ρ(X)=1 to so-called the group of Atkin–Lehner involutions. This was expected in Hosono, Lian, Oguiso and Yau's paper.