A topological approach to Soergel theory

Roman Bezrukavnikov (1) , Simon Riche (2, 3)

1 MIT - Massachusetts Institute of Technology
2 LMBP - Laboratoire de Mathématiques Blaise Pascal
3 UCA [2017-2020] - Université Clermont Auvergne [2017-2020]

We develop a "Soergel theory" for Bruhat-constructible perverse sheaves on the flag variety G/B of a complex reductive group G, with coefficients in an arbitrary field k. 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 T-monodromic k-sheaves on G/U (where U,T ⊂ B are the unipotent radical and the maximal torus), as a monoidal category.

Domaines: Théorie des représentations [math.RT]

Roman Bezrukavnikov, Simon Riche. A topological approach to Soergel theory. Trends in Mathematics, Birkhäuser/Springer, pp.267-343, 2022, Representation theory and algebraic geometry—a conference celebrating the birthdays of Sasha Beilinson and Victor Ginzburg. ⟨hal-01846873⟩