arXiv subjects

Simon Riche

Publications and source records attributed to Simon Riche.

60 records · Page 4Linked to original sources

arXiv2011

Affine braid group actions on derived categories of Springer resolutions

In this paper we construct and study an action of the affine braid group associated to a semi-simple algebraic group on derived categories of coherent sheaves on various varieties related to the Springer resolution of the nilpotent cone. In particular, we describe explicitly the action of the Artin braid group. This action is a "categorical version" of Kazhdan--Lusztig--Ginzburg's construction of the affine Hecke algebra, and is used in particular by the first author and Ivan Mirkovic in the course of the proof of Lusztig's conjectures on equivariant K-theory of Springer fibers.

math.RT↗

arXiv2011

Koszul duality and semisimplicity of Frobenius

A fundamental result of Beilinson-Ginzburg-Soergel states that on flag varieties and related spaces, a certain modified version of the category of l-adic perverse sheaves exhibits a phenomenon known as Koszul duality. The modification essentially consists of discarding objects whose stalks carry a nonsemisimple action of Frobenius. In this paper, we prove that a number of common sheaf functors (various pull-backs and push-forwards) induce corresponding functors on the modified category or its triangulated analogue. In particular, we show that these functors preserve semisimplicity of the Frobenius action.

math.RT↗

arXiv2011

Constructible sheaves on affine Grassmannians and geometry of the dual nilpotent cone

In this paper we study the derived category of sheaves on the affine Grassmannian of a complex reductive group G, contructible with respect to the stratification by G(C[[x]])-orbits. Following ideas of Ginzburg and Arkhipov-Bezrukavnikov-Ginzburg, we describe this category (and a mixed version) in terms of coherent sheaves on the nilpotent cone of its Langlands dual reductive group. We also show, in the mixed case, that restriction to the nilpotent cone of a Levi subgroup corresponds to hyperbolic localization on affine Grassmannians.

math.RT↗

arXiv2010

Koszul duality and Frobenius structure for restricted enveloping algebras

Let g be the Lie algebra of a connected, simply connected semisimple algebraic group over an algebraically closed field of sufficiently large positive characteristic. We study the compatibility between the Koszul grading on the restricted enveloping algebra (Ug)_0 of g constructed in a previous paper, and the structure of Frobenius algebra of (Ug)_0. This answers a question raised to the author by W. Soergel.

math.RT↗

arXiv2010

Koszul duality and modular representations of semi-simple Lie algebras

In this paper we prove that if G is a connected, simply-connected, semi-simple algebraic group over an algebraically closed field of sufficiently large characteristic, then all the blocks of the restricted enveloping algebra (Ug)_0 of the Lie algebra g of G can be endowed with a Koszul grading (extending results of Andersen, Jantzen and Soergel). We also give information about the Koszul dual rings. Our main tool is the localization theory in positive characteristic developed by Bezrukavnikov, Mirkovic and Rumynin.

math.RT↗

arXiv2009

Linear Koszul Duality

In this paper we construct, for F_1 and F_2 subbundles of a vector bundle E, a "Koszul duality" equivalence between derived categories of G_m-equivariant coherent (dg-)sheaves on the derived intersection of F_1 and F_2 inside E, and the corresponding derived intersection of the orthogonals of F_1 and F_2 inside the dual vector bundle E^*. We also propose applications to Hecke algebras.

math.RT↗