  An integral hyperbolic lattice is called reflective if its automorphism group
is generated by reflections, up to finite index. Since 1981, it is known that
their number is essentially finite.
  We show that K3 surfaces over C with reflective Picard lattices can be
characterized in terms of compositions of their self-correspondences via moduli
of sheaves with primitive isotropic Mukai vector: Their self-correspondences
with integral action on the Picard lattice are numerically equivalent to
compositions of a finite number of especially simple self-correspondences via
moduli of sheaves.
  This relates two topics: Self-correspondences of K3 surfaces via moduli of
sheaves and Arithmetic hyperbolic reflection groups. It also raises several
natural unsolved related problems.
