Cohomology of tilting modules over quantum groups and t-structures on derived categories of coherent sheaves

By R. Bezrukavnikov. 2004

75 citations recorded by the source corpus.

The paper is concerned with cohomology of the small quantum group at a root of unity, and of its upper triangular subalgebra, with coefficients in a tilting module. It turns out to be related to irreducible objects in the heart of a certain t-structure on the derived category of equivariant coherent sheaves on the Springer resolution, and to equivariant coherent IC sheaves on the nil-cone. The support of the cohomology is described in terms of cells in affine Weyl groups. The basis in the Grothendieck group provided by the cohomology modules is shown to coincide with the Kazhdan-Lusztig basis, as predicted by J. Humphreys and V. Ostrik.The proof is based on the results of [ABG ], [AB] and [B], which allow us to reduce the question to purity of IC sheaves on affine flag varieties.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2d17cca2e5a7b7155e8698ba61b47a9f69f0828b

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On two geometric realizations of an affine Hecke algebra (2012, background): “[16], [12]) the following simpler version is sufficient.”
Quantum groups, the loop Grassmannian, and the Springer resolution (2003, result): “eover, an additional argument based on results of [AB] shows that the parameters labelling the tilting objects in block(U) and the simple perverse coherent sheaves on Ne correspond to each other, see [B5]. It follows, in particular, that the support of the quantum group cohomology of a tilting U-mod”
Equivariant Satake category and Kostant-Whittaker reduction (2007, unknown): “Corollary 1. a) The derived graded Harish-Chandra bimodule category is Koszul dual to a graded version of the loop-rotation equivariant Satake category, i.e. (cf. [7] , [2] , [10] ) there exists a functor Φ : iii) The image of Φ generates the target category as a triangulated category.”
Gale duality and Koszul duality (2008, background): “(See [ Bez06 ] for definitions of exceptional collections and mutations.) We leave the proof as an exercise to the reader.”
Perverse coherent sheaves (2009, background): “See [4], [5] for more information on this example.”
Koszul duality for Kac–Moody groups and characters of tilting modules (2017, background): “ : KbA → T whose restriction to A is the inclusion functor. In addition, if A is the heart of a t-structure (and hence an abelian category), this functor factors through a functor real : DbA → T . In [Be], this result is only stated in the case where A is the heart of a t-structure. For details in a”