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We consider certain universal functors on symmetric quotient stacks ofAbelian varieties. In dimension two, we discover a family of$\mathbb{P}$-functors which induce new derived autoequivalences of Hilbertschemes of points on Abelian surfaces; a set of braid relations on aholomorphic symplectic sixfold; and a pair of spherical functors on the Hilbertsquare of an Abelian surface, whose twists are related to the well-known Horjatwist. In dimension one, our universal functors are fully faithful, giving riseto a semiorthogonal decomposition for the symmetric quotient stack of anelliptic curve (which we compare to the one discovered by Polishchuk--Van denBergh), and they lift to spherical functors on the canonical cover, inducingtwists which descend to give new derived autoequivalences here as well.

Universal functors on symmetric quotient stacks of Abelian varieties