Groups actions, and D equivalences of categories of coherent sheaves of symplectic resolutions

Abstract: Let k be an algebraically closed field of characteristic p>>0. Let $X\rightarrow Y$ be a symplectic resolution. There are two questions which motivates this work. One question is a construction of an action of a group on the category $\mathcal{C}:=D^b(Coh(X))$ - The bounded derived category of coherent sheaves of the symplectic resolution X. Second question is understanding equivalence functors between derived categories of coherent sheaves for different symplectic resolutions of Y. Let G/k be a reductive group. In this paper, we construct a local system on a topological space called $V^0_{\mathbb{C}}$ with value the category $D^b(Coh(T^*G/P))$ for a parabolic subgroup P. This induces an action of $\pi_1 V^0_{\mathbb{C}}$ on the category. In another paper we further explain how a refinement of this local system construction, gives an answer to the second question, showing that these equivalence functors, are parametrized by homotopy classes of maps between certain points in the base space. We also lift the result to characteristic zero.