Groupes et algébres de lie

(1.1) This paper concerns three aspects of the action of a compact group K on a space X . The ®rst is concrete and the others are rather abstract. (1) Equivariantly formal spaces. These have the property that their cohomology may be computed from the structure of the zero and one dimensional orbits of the action of a maximal torus in K. (2) Koszul duality. This enables one to translate facts about equivariant cohomology into facts about its ordinary cohomology, and back. (3) Equivariant derived category. Many of the results in this paper apply not only to equivariant cohomology, but also to equivariant intersection cohomology. The equivariant derived category provides a framework in both of these may be considered simultaneously, as examples of ``equivariant sheaves''. We treat singular spaces on an equal footing with nonsingular ones. Along the way, we give a description of equivariant homology and equivariant intersection homology in terms of equivariant geometric cycles. Most of the themes in this paper have been considered by other authors in some context. In Sect. 1.7 we sketch the precursors that we know about. For most of the constructions in this paper, we consider an action of a compact connected Lie group K on a space X , however for the purposes of the introduction we will take K  S1 to be a torus. Invent. math. 131, 25±83 (1998)

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Equivariant cohomology, Koszul duality, and the localization theorem

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

The classical McKay correspondence relates representations of a finite subgroup
G ⊂ SL(2,C) to the cohomology of the well-known minimal resolution of the
Kleinian singularity C2/G. Gonzalez-Sprinberg and Verdier [10] interpreted the
McKay correspondence as an isomorphism on K theory, observing that the representation
ring of G is equal to the G-equivariant K theory of C2. More precisely,
they identify a basis of the K theory of the resolution consisting of the classes of
certain tautological sheaves associated to the irreducible representations of G.

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The McKay correspondence as an equivalence of derived categories

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety $X$. The motivation for this is M. Kontsevich's homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dim $X\geq 2$, our braid group actions are always faithful. We describe conjectural mirror symmetries between smoothings and resolutions of singularities which lead us to find examples of braid group actions arising from crepant resolutions of various singularities. Relations with the McKay correspondence and with exceptional sheaves on Fano manifolds are given. Moreover, the case of an elliptic curve is worked out in some detail.

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Braid group actions on derived categories of coherent sheaves

Deformation theory of objects in homotopy and derived categories III: abelian categories

0.1. This paper arose from an attempt to solve the following problem. Let (g, K) be a Harish-Chandra pair, i.e. g is a complex reductive Lie algebra, and K is an algebraic group with an action K -Aut(g) and an embedding k = Lie K c g, satisfying some standard conditions (see 1.1 below). Let Z be the center of the enveloping algebra U(g) . Fix a regular character 0: Z -C. Let ((g, K) be the category of (g, K)-modules and t,(g, K) c l((g, K) the subcategory consisting of modules annihilated by Ker 0 . Then by the localization theorem this category X (g, K) can be described geometrically. Namely, fix a Borel subalgebra b c g and a dominant weight A corresponding to 0. Consider the algebra D, of twisted differential operators on the flag space X of g. Then X4/g, K) = J((DZ, K), the K-equivariant D,-modules on X. This result allows us to study many properties of Harish-Chandra modules geometrically. But it does not give a geometric interpretation of Ext-groups of modules in t, (g, K). Namely, let M, N E 4t (gi, K). From the point of view of representation theory the interesting objects are Ext.,, (, K) (M, N), the Ext-groups in the category of all (g, K)-modules. But these Ext's do not admit localization since arbitrary (g, K)-modules do not localize. Our main result is a geometric interpretation of these Ext-groups and, more precisely, of the corresponding derived category. Let us describe it. Let to(g, K) c l((g, K) be the subcategory of 0-finite modules. That is, each element m of M E Ito(g, K) is annihilated by some power of Ker 0. Recall the localization for the category to(g, K) (precise definitions will be given later). Let G be the algebraic group of automorphisms of g, H c G a maximal torus, [ = Lie H. The flag variety X has a natural H-monodromic structure X -X. Let At(Dk) denote the category of weakly H-equivariant Dk-modules. Elements of 1#(Dk) are called monodromic D-modules on X.

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Localization for derived categories of (,)-modules

We introduce the notion of (homological) G-smoothness for a complex G-variety X, where G is a connected affine algebraic group. This is based on the notion of smoothness for dg algebras and uses a suitable enhancement of the G-equivariant derived category of X. If there are only finitely many G-orbits and all stabilizers are connected, we show that X is G-smooth if and only if all orbits O satisfy H^*(O; R)=R. On the way we prove several results concerning smoothness of dg categories over a graded commutative dg ring.

Smoothness of equivariant derived categories

We introduce new enhancements for the bounded derived category $D^b(Coh(X))$ of coherent sheaves on a suitable scheme $X$ and for its subcategory $Perf(X)$ of perfect complexes. They are used for translating Fourier-Mukai functors to functors between derived categories of dg algebras, for relating homological smoothness of $Perf(X)$ to geometric smoothness of $X,$ and for proving homological smoothness of $D^b(Coh(X)).$ Moreover, we characterize properness of $Perf(X)$ and $D^b(Coh(X))$ geometrically.

Valery A. Lunts

Papers

Deformation theory of objects in homotopy and derived categories III: abelian categories

Localization for derived categories of (,)-modules

Smoothness of equivariant derived categories

New enhancements of derived categories of coherent sheaves and applications

Smoothness of equivariant derived categories