P-functor versions of the Nakajima operators

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For every smooth quasi-projective surface X we construct a series of Pⁿ⁻¹-functors Hₗ,ₙ: D(X x X[l]) --> D(X[n+l]) between the derived categories of the Hilbert schemes of points for n>maxl,1 using the derived McKay correspondence. They can be considered as analogues of the Nakajima operators. The functors also restrict to Pⁿ⁻¹-functors on the generalised Kummer varieties. We also study the induced autoequivalences and obtain, for example, a universal braid relation in the groups of derived autoequivalences of Hilbert squares of K3 surfaces and Kummer fourfolds.