Symplectic involutions of $K{3}^{[n]}$ type and Kummer $n$ type manifolds

In this paper we describe the fixed locus of a symplectic involution on a hyperk"ahler manifold of type $K{3}^{[n]}$ or of Kummer $n$ type. We prove that the fixed locus consists of finitely many copies of Hilbert schemes of $K3$ surfaces of lower dimensions and isolated fixed points.