Perverse Motives and Graded Derived Category

Abstract

For a variety with a Whitney stratification by affine spaces, we study categories of motivic sheaves which are constant mixed Tate along the strata. We are particularly interested in those cases where the category of mixed Tate motives over a point is equivalent to the category of finitedimensional bigraded vector spaces. Examples of such situations include rational motives on varieties over finite fields and modules over the spectrum representing the semisimplification of de Rham cohomology for varieties over the complex numbers. We show that our categories of stratified mixed Tate motives have a natural weight structure. Under an additional assumption of pointwise purity for objects of the heart, tilting gives an equivalence between stratified mixed Tate sheaves and the bounded homotopy category of the heart of the weight structure. Specializing to the case of flag varieties, we find natural geometric interpretations of graded category O and Koszul duality.

Some geometric facets of the Langlands correspondence for real groups

“…[3, § 4.5]). Soergel has explained to me how combining the arguments of this note with a joint project of his and Wendt's on 'motivic representation theory' (see [17]) should allow this basic idea to be carried through (also see below Acknowledgements). This perspective is also explicit in [2, § G; 3, § 4].…”

The Intersection Motive of the Moduli Stack of Shtukas

“…By definition, DTM(S) is the subcategory generated by motives of P n S (n ≥ 0) and their duals. Soergel and Wendt [SW18] have extended Levine's observation to the case when X is an S-scheme equipped with a so-called cellular Whitney-Tate stratification: loosely speaking, this condition means that the strata of X are built out of products of G m,S or A 1 S , and that one needs to be able to control the singularities of the closures of the strata. While this condition is rather restrictive in general, it turns out that several varieties X of interest in geometric representation theory do carry such stratifications.…”

“…While this condition is rather restrictive in general, it turns out that several varieties X of interest in geometric representation theory do carry such stratifications. For example, the flag variety X = G/B associated with a split reductive k-group G and a Borel subgroup B, equipped with its stratification by B-orbits, has this property [SW18,Prop. 4.10].…”

“…Given i), the construction in ii) ultimately boils down (cf. §5, §6) to an extension of the methods of [SW18] and of Soergel-Virk-Wendt [SVW18] from ordinary schemes with an action of an algebraic group to ind-schemes with an action of a pro-algebraic group. Our constructions are also applied in forthcoming work [RS19] to construct a motivic Satake equivalence in this set-up.…”

“…Definition 3.1.8. Following Soergel-Wendt [SW18,Def. 4.5] (who consider the case of finite type schemes over fields), the stratification ι : X + → X is called Whitney-Tate if ι * ι * 1 X + ∈ DTM(X + ), i.e., (ι| Xv ) * (ι| Xw ) * 1 Xw is a Tate motive on X v for each v, w ∈ W .…”