In this talk we consider auto-equivalences of the bounded derived category D(X) of coherent sheaves on a smooth projective complex variety X. By a result of Orlov, any such auto-equivalence induces an (ungraded) automorphism of the singular cohomology H(X,\Q). If X is a K3 surface, then work of Mukai, Orlov, Huybrechts, Macr and Stellari completely describes the image of the map \rho_X : \Aut D(X) --> Aut(H(X, \Q)). We will study the image of \rho_X for higher-dimensional hyperkhler varieties. An important tool is a certain Lie algebra acting on H(X, \Q), introduced by Verbitsky, Looijenga and Lunts. We show that this Lie algebra is a derived invariant, and use this to study the image of \rho_X.

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