Let X be a K3 surface and let \({{\,\textrm{Spl}\,}}(r;c_1,c_2)\) be the moduli space of simple sheaves on X of fixed rank r and Chern classes \(c_1\) and \(c_2\) . Under suitable assumptions, to a pair (F, W) (respectively, (F, V)) where \(F\in {{\,\textrm{Spl}\,}}(r;c_1,c_2)\) and \(W\subset H^0(F)\) (resp. \(V^*\subset H^1(F^*)\) ) is a vector subspace, we associate a simple syzygy bundle (resp. extension bundle) on X. We show that both syzygy bundles and extension bundles can be constructed in families and that the induced morphism to a different component of the moduli of simple sheaves is a locally closed embedding. We show that this construction associates with every Lagrangian (resp. isotropic) algebraic subspace of \({{\,\textrm{Spl}\,}}(r;c_1,c_2)\) an induced Lagrangian (resp. isotropic) algebraic subspace of a different component of the moduli of simple sheaves.
