Lang map (${\mathcal{L}}_F(g)=g^{-1}F(g)$)

For the Lang map, ${\mathcal{L}}_F(gh)=h^{-1}{\mathcal{L}}_F(g)F(h)$. This twisted right-translation identity determines which subgroups act on its inverse images. Fibres are left $G^F$-cosets, by the Lang theorem for algebraic groups.