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Noncommutative counterparts of the Springer resolution. (English) Zbl 1135.17011
Sanz-Solé, Marta (ed.) et al., Proceedings of the international congress of mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-022-7/hbk). 1119-1144 (2006).
The Springer resolution given by the moment map from the cotangent bundle $$\widetilde{\mathcal{N}}$$ of the flag variety of a connected semisimple algebraic group of adjoint type to the variety $$\mathcal{N}$$ of nilpotent elements of its Lie algebra plays an important role in geometric representation theory. Recently, non-commutative resolutions of $$\mathcal{N}$$, which are uniquely determined by the so-called exotic $$t$$-structure on the bounded derived category $$D^b(\widetilde{\mathcal{N}})$$ of equivariant coherent sheaves on $$\widetilde{\mathcal{N}}$$, have appeared in several representation-theoretic and algebro-geometric constructions. The paper under review gives a survey of these constructions mainly due to the author and his collabarators.
In particular, their applications in a local version of the geometric Langlands duality program and the categorification of the representation theory of affine Hecke algebras or in a derived localization theorem for representations of Lie algebras of semisimple algebraic groups in prime characteristic are given in considerable detail. Related topics as describing the cohomology of tilting modules for quantized universal enveloping algebras at roots of unity by equivariant exotic sheaves, i.e., the objects of the heart of the exotic $$t$$-structure on $$D^b(\widetilde{\mathcal{N}})$$, Bridgeland’s theory of stability conditions on triangulated categories, and the construction of non-commutative resolutions of symplectic quotient singularities which yield a special case of the categorical McKay correspondence are mentioned as well.
For the entire collection see [Zbl 1095.00005].

##### MSC:
 17B50 Modular Lie (super)algebras 18F99 Categories in geometry and topology 20G05 Representation theory for linear algebraic groups 17B37 Quantum groups (quantized enveloping algebras) and related deformations 20G42 Quantum groups (quantized function algebras) and their representations 22E67 Loop groups and related constructions, group-theoretic treatment
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