For generic K3 surfaces, a symmetry of the Hilbert scheme of points is induced by a derived autoequivalence if and only if it acts trivially on the discriminant group of the second cohomology lattice.
Birational geometry of singular moduli spaces of O’Grady type
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Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces
For generic K3 surfaces, a symmetry of the Hilbert scheme of points is induced by a derived autoequivalence if and only if it acts trivially on the discriminant group of the second cohomology lattice.