Affine braid group actions on derived categories of Springer resolutions

Abstract

Abstract. In this paper we construct and study an action of the affine braid group associated to a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a "categorical version" of Kazhdan-Lusztig-Ginzburg's construction of the affine Hecke algebra, and is used in particular in [BM] in the course of the proof of Lusztig's conjectures on equivariant K-theory of Springer fibers.

Kostant section, universal centralizer, and a modular derived Satake equivalence

“…Hence, using [BR,Lemma 1.4.1(1)], to prove that (4.5.2) is an isomorphism it suffices to prove that it becomes an isomorphism after applying F ⊗ R (−) for all geometric points F of R. The latter fact was proved in Proposition 3.5.5(1).…”

“…Since both group schemes are closed subschemes of G R × Spec(R) S R , and since this morphism is compatible with their inclusion in G R × Spec(R) S R , (4.5.3) must be a closed embedding. Since the R-algebra O( S R × S R I R S ) is of finite type and flat over R, and since I R S is also R-flat, by [BR,Lemma 1.4.1(1)], to prove that (4.5.3) is an isomorphism it suffices to prove that, for any geometric point F of R, the morphism…”

“…To prove that this morphism is an isomorphism it suffices to prove that its cone is isomorphic to 0. Since both I R S and Ω S R are flat over R, to prove this it suffices to prove that for any geometric point F of R, the induced morphism F ⊗ R I R S → F ⊗ R Ω S R is an isomorphism (see [BR,Lemma 1.4.1(1)]). However, by Lemma 4.4.3(3), F ⊗ R I R S is the Lie algebra I S of §3.4 for the group G F , hence the desired claim follows from Theorem 3.4.2.…”

“…However by Grothendieck's vanishing theorem [Ha,Theorem III.2.7] this complex is bounded, and since X is projective over A its cohomology sheaves are finitely generated over A (see [Ha,Theorem III.5.2]). Hence by [BR,Lemma 1.4.1(2)] it suffices to prove that for any geometric point F of R, the complex F ⊗ L R RΓ(X, I) is concentrated in degree 0. Now one can easily check that we have…”

Categorical action of the extended braid group of affine type A

“…Categorical actions of Rouquier in type ADE are known to be faithful by a result of Brav and Thomas [BT11]. In addition Riche and Bezuriakhov [Ric08,BR12] constructed a faithful action of the affine braid group on a derived category of coherent sheaves. For all we know most of the results of faithfulness rely on the result of Khovanov and Seidel. Looking at categorical actions of the whole mapping class groups of an oriented surface with boundary components, the situation is drastically different, there is essentially only one such action given by Lipschitz-OzsvathSzabo in the context of bordered Heegaard-Floer homology and the proof of faithfulness in their setting is the same in spirit as the one of Khovanov and Seidel and as in this paper: the dimensions of the space of morphisms count certain minimal intersection numbers.…”