  Components of the Moduli space of sheaves on a K3 surface are parametrized by
a lattice; the (algebraic) Mukai lattice. Isometries of the Mukai lattice often
lift to symplectic birational isomorphisms of the collection of components. An
example of such a birational isomorphism is the Abel-Jacobi map relating the
Hilbert scheme of g points on a K3 of degree 2g-2 to an integrable system: the
union of Jacobians of hyperplane sections (curves) of genus g. The main results
are:
  1) We construct a stratified version of a Mukai elementary transformation
modeled after dual pairs of Springer resolutions of nilpotent orbits. It
applies to a holomorphic-symplectic variety M with a stratification where the
first stratum is a P^n bundle, but lower strata are Grassmannian bundles. The
resulting (transformed) symplectic variety W admits a stratification by the
dual Grassmannian bundles.
  2) The group of reflections of the Mukai lattice, which act trivially on the
second cohomology of the K3 surface, acts on moduli spaces of sheaves (with
``minimal'' first Chern class) as birational stratified elementary
transformations.
  3) We derive a Picard-Lefschetz type formula identifying the isomorphism of
cohomology rings of a holomorphic-symplectic variety M and its stratified
transform W as the cup product with an algebraic correspondence.
