Non-natural non-symplectic involutions on symplectic manifolds of K3[²]-type

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We study non-symplectic involutions on irreducible symplectic manifolds of K3[²]-type with 19 parameters, which is the second largest possible. We classify the conjugacy classes of cohomological representations into four different types and show that there are at most five deformation types, two of which are given by natural involutions and their flops. Next, we give a geometric realisation of one of the new types using moduli spaces of sheaves on K3 surfaces. The geometry of the manifold and the new involution is described in detail.