We construct an equivalence of categories from a strong categorical sl(2) action, following the work of Chuang-Rouquier. As an application, we give an explicit, natural equivalence between the derived categories of coherent sheaves on cotangent bundles to complementary Grassmannians. Date: July 1, 2011. 1 2 SABIN CAUTIS, JOEL KAMNITZER, AND ANTHONY LICATA 6.4. The equivalence 33 6.5. Mukai flops 33 6.6. The case of T ⋆ G(2, 4) 33 References 34 where the sum is finite since V (λ) = 0 for λ ≫ 0.

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... e can lift f (λ+s) e (s) to the composition of functors F (λ+s) • E (s) . We then form a complex of functors Θ * whose terms are (the · represents a shift in the grading). The connecting maps of the complex are defined via various adjunction morhisms (section 2.3). The main result of this paper, Theorem 2.8, states that if our categories D(λ) are triangulated then the convolution of this complex is an equivalence T : 1.3. Application to stratified Mukai flops. Our main application in this paper is to construct equivalences between derived categories of coherent sheaves on cotangent bundles of Grassmannians. We fix a positive integer N and let D(λ) := DCoh(T ⋆ G(k, N )) be the bounded derived category of coherent sheaves on the cotangent bundle of the Grassmannian where λ = N − 2k. In [CKL2] we constructed a strong categorical sl 2 action on these categories, where the functors E, F act by natural correspondences. Hence by the main result of this paper, we get a non-trivial, natural equivalence (see Corollary 6.2). When k = 1, T ⋆ G(1, N ) and T ⋆ G(N − 1, N ) differ by a standard Mukai flop and the equivalence we produce is already known (see [Ka1] or [N1]). When k > 1 the varieties T ⋆ G(k, N ) and T ⋆ G(N − k, N ) are related by a stratified Mukai flop. The problem of constructing such an equivalence (3) in these cases was posed by Namikawa in [N2]. Our work gives the first solution to this problem with the exception of the case G(2, 4) where an equivalence was constructed by Kawamata in [Ka2] . We should note that the idea of using categorical sl 2 actions to construct the equivalence (3) was first proposed by Rouquier in [R1, section 4.4.2]. Despite the indirect nature of our construction, we are able to give a fairly concrete description of the equivalence, which we do in section 6. In particular we prove that it is induced by a Fourier-Mukai kernel which is a Cohen-Macauley sheaf. Similar geometric settings for equivalences constructed from strong categorical sl 2 actions, including convolutions of affine Schubert varieties and Nakajima quiver varieties, are considered in [CKL1] and [CKL3], respectively. academic year. Equivalences from strong categorical sl 2 actions In this section we begin by reviewing the concept of a strong categorical sl 2 action. Then we describe how to form a complex of functors from such an action. Finally we state our result (Theorem 2.8) which explains when and how this complex of functors gives an equivalence of categories.