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2022 | OriginalPaper | Chapter

A Topological Approach to Soergel Theory

Authors : Roman Bezrukavnikov, Simon Riche

Published in: Representation Theory and Algebraic Geometry

Publisher: Springer International Publishing

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Abstract

We develop a “Soergel theory” for Bruhat-constructible perverse sheaves on the flag variety GB of a complex reductive group G, with coefficients in an arbitrary field https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq1_HTML.gif . Namely, we describe the endomorphisms of the projective cover of the skyscraper sheaf in terms of a “multiplicative” coinvariant algebra and then establish an equivalence of categories between projective (or tilting) objects in this category and a certain category of “Soergel modules” over this algebra. We also obtain a description of the derived category of unipotently T monodromic https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq2_HTML.gif sheaves on GU (where U, T ⊂ B are the unipotent radical and the maximal torus), as a monoidal category, in terms of coherent sheaves on the formal neighborhood of the base point in https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq3_HTML.gif , where https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq4_HTML.gif is the https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq5_HTML.gif -torus dual to T.

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Footnotes
1
All the torsors we will encounter in the present paper will be locally trivial for the Zariski topology.
 
2
All our pro-objects are tacitly parametrized by \(\mathbb {Z}_{\geq 0}\) (with its standard order).
 
3
This assumption is probably unnecessary. But since this is the setting we are mostly interested in, we will not consider the possible extension of this claim to the characteristic-0 setting.
 
4
Recall that in the étale setting, the U-equivariant and B-constructible derived categories are different if p > 0, due to the existence of nonconstant local systems on affine spaces. Here https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq1144_HTML.gif is the full triangulated subcategory of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-82007-7_7/449373_1_En_7_IEq1145_HTML.gif generated by pushforwards of constant local systems on strata.
 
5
Namely, it is claimed in this proof that the complex denoted “C” is concentrated in positive perverse degrees. But the arguments given there only imply that its negative perverse cohomology objects vanish.
 
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Metadata
Title
A Topological Approach to Soergel Theory
Authors
Roman Bezrukavnikov
Simon Riche
Copyright Year
2022
DOI
https://doi.org/10.1007/978-3-030-82007-7_7

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