Derived category D G (X) and functors.- DG-modules and equivariant cohomology.- Equivariant cohomology of toric varieties.

We construct categories of Harish-Chandra bimodules for affine Lie algebras analogous to Harish-Chandra bimodules with infinitesimal characters for simple Lie algebras, addressing an old problem raised by I. Frenkel and Malikov. Under an integrality hypothesis, we establish monoidal equivalences between blocks of our affine Harish-Chandra bimodules and versions of the affine Hecke category. To do so, we introduce a new singular localization onto Whittaker flag manifolds. The argument for the latter specializes to a somewhat alternative treatment of regular localization, and identifies the category of Lie algebra representations with a generalized infinitesimal character as the parabolic induction, in the sense of categorical representation theory, of a Steinberg module.

Equivariant Sheaves and Functors

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

We continue our study of the monoidal category $D(G)$. At the level of cohomology we transfer the duality functor to the derived category of Hecke dg-modules. In the process we develop a more general and streamlined approach to the anti-involution first defined by Ollivier and Schneider. We also verify that the tensor product on $D(G)$ corresponds to an operadic tensor product on the dg-side. This uses a result of Schn\"{u}rer on dg-categories with a model structure.

dg-Hecke duality and tensor products

The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) $G$-module, where $G$-modules are structures which differentiate to representations. Many geometric structures involving Lie groupoids and stacks are classified by the cohomology of sheaves taking values in $G$-modules and not in representations, including $S^1$-groupoid extensions and equivariant gerbes. Examples of such sheaves are $\mathcal{O}^*$ and $\mathcal{O}^*(*D)\,,$ where the latter is the sheaf of invertible meromorphic functions with poles along a divisor $D\,.$ We show that there is an infinitesimal description of $G$-modules and a corresponding Lie algebroid cohomology. We then define a generalized van Est map relating these Lie groupoid and Lie algebroid cohomologies, and study its kernel and image. Applications include the integration of several infinitesimal geometric structures, including Lie algebroid extensions, Lie algebroid actions on gerbes, and certain Lie $\infty$-algebroids.

http://academic.oup.com/imrn/advance-article-pdf/doi/10.1093/imrn/rnab027/37038882/rnab027.pdf

Cohomology of Lie Groupoid Modules and the Generalized van Est Map

We show that representations of convolution algebras such as Lustzig's graded affine Hecke algebra or the quiver Hecke algebra and quiver Schur algebra in (affine) type A can be realised in terms of certain equivariant motivic sheaves called Springer motives. To this end, we lay foundations to a motivic Springer theory and prove formality results using weight structures. 
As byproduct, we express Koszul and Ringel duality in terms of a weight complex functor and show that partial quiver flag varieties in affine type A (with cyclic orientation) admit an affine paving.

Motivic Springer Theory

This paper studies abelian categories that can be decomposed into smaller abelian categories via iterated recollements - such a decomposition we call a stratification. Examples include the categories of (equivariant) perverse sheaves and epsilon-stratified categories (in particular highest weight categories) in the sense of Brundan-Stroppel (2018). We give necessary and sufficient conditions for an abelian category with a stratification to be equivalent to a category of finite dimensional modules of a finite dimensional algebra - this generalizes the main result of Cipriani-Woolf (2022). Furthermore, we give necessary and sufficient conditions for such a category to be epsilon-stratified - this generalizes the characterisation of highest weight categories given by Krause (2017).

Equivariant Sheaves and Functors

Citations

Affine Harish-Chandra bimodules and Steinberg--Whittaker localization

dg-Hecke duality and tensor products

Cohomology of Lie Groupoid Modules and the Generalized van Est Map

Motivic Springer Theory

Stratifications of abelian categories

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