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Wolfgang Soergel

Publications and source records attributed to Wolfgang Soergel.

10 recordsLinked to original sources
arXiv2018

Equivariant motives and geometric representation theory. (with an appendix by F. Hörmann and M. Wendt)

We consider categories of equivariant mixed Tate motives, where equivariant is understood in the sense of Borel. We give the two usual definitions of equivariant motives, via the simplicial Borel construction and via algebraic approximations of it. The definitions turn out to be equivalent and give rise to a full six-functor formalism. For rational étale motives over a finite field or the homotopical stable algebraic derivator arising from the semisimplified Hodge realization, the equivariant mixed Tate motives provide a graded version of the equivariant derived category. We show that, in sufficiently nice and clean cases, these categories admit weight structures; moreover, a tilting result holds which identifies the category of equivariant mixed Tate motives with the bounded homotopy category of the heart of its weight structure. This can be seen as a formality result for equivariant derived categories. We also discuss convolution functors on equivariant mixed Tate motives, and consequences for the categorification of the Hecke algebra and some of its modules.

math.RT↗
arXiv2015

Perverse motives and graded derived category $\mathcal{O}$

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 finite-dimensional 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 $\mathcal O$ and Koszul duality.

math.RT↗
arXiv2014

Proper base change for separated locally proper maps

We introduce and study the notion of a locally proper map between topological spaces. We show that fundamental constructions of sheaf theory, more precisely proper base change, projection formula, and Verdier duality, can be extended from continuous maps between locally compact Hausdorff spaces to separated locally proper maps between arbitrary topological spaces.

math.AT↗
arXiv2012

Modulare Koszul-Dualit"at

We prove an analogon of Koszul duality for category O in positive characteristic. However, there are no Koszul rings, and we do not prove an analog of the Kazhdan-Lusztig conjectures in this context.

math.RT↗
arXiv2012

Modular Koszul duality

We prove an analogue of Koszul duality for category $\mathcal{O}$ of a reductive group $G$ in positive characteristic $\ell$ larger than 1 plus the number of roots of $G$. However there are no Koszul rings, and we do not prove an analogue of the Kazhdan--Lusztig conjectures in this context. The main technical result is the formality of the dg-algebra of extensions of parity sheaves on the flag variety if the characteristic of the coefficients is at least the number of roots of $G$ plus 2.

math.RT↗
arXiv2008

Herleitung von Skalarprodukten aus Symmetrieprinzipien

This is an attempt to model ambient space as a three-dimensional real affine space with a distinguished group of automorphisms containing the translations and acting freely and transitively on pairs consisting of a half-plane together with a half-line on its boundary. From there the existence of an invariant scalar product is deduced, which then also implies Pythagoras theorem in a quite precise form. This is in contrast to the usual procedure to model ambient space by asking for a distinguished scalar product and using Pythagoras theorem as known from high school to connect with reality.

math.HO↗
arXiv2007

Andersen Filtration and Hard Lefschetz

On the space of homomorphisms from a Verma module to an indecomposable tilting module of the BGG-category O we define a natural filtration following Andersen and establish a formula expressing the dimensions of the filtration steps in terms of coefficients of Kazhdan-Lusztig polynomials.

math.RT↗
arXiv2006

Andersen-Filtrierung und harter Lefschetz

We consider the principal block of category O and its graded version. On the space of homomorphisms from a Verma module to an indecomposable tilting module we may define natural filtrations following Andersen. The arguments given in this article prove that these filtrations are compatible with the graded structure, although explicitely we only show that the dimensions of the sucessive subquotients of the filtration are compatible with this intuition. This statement is very similar to the semisimplicity of the subquotients of the Jantzen filtration proved by Beilinson and Bernstein, but the method of proof is quite different. I would like to know how to directly relate both results, as this would give an alternative proof of said semisimplicity.

math.RT↗