The parabolic exotic t-structure

Abstract

International audience Let G be a connected reductive algebraic group over an algebraically closed field k, with simply connected derived subgroup. The exotic t-structure on the cotangent bundle of its flag variety T^*(G/B), originally introduced by Bezrukavnikov, has been a key tool for a number of major results in geometric representation theory, including the proof of the graded Finkelberg-Mirkovic conjecture. In this paper, we study (under mild technical assumptions) an analogous t-structure on the cotangent bundle of a partial flag variety T^*(G/P). As an application, we prove a parabolic analogue of the Arkhipov-Bezrukavnikov-Ginzburg equivalence. When the characteristic of k is larger than the Coxeter number, we deduce an analogue of the graded Finkelberg-Mirkovic conjecture for some singular blocks. Soit G un groupe algébrique réductif connexe sur un corps k algébriquement clos. La t-structure exotique sur le fibré cotangent de sa variété de drapeaux T^*(G/B), introduite à l'origine par Bezrukavnikov, a été un outil clé pour de nombreux résultats majeurs en théorie géométrique des représentations, en particulier la démonstration de la conjecture de Finkelberg-Mirkovic graduée. Dans cet article, nous étudions (sous de légères hypothèses techniques) une t-structure analogue sur le fibré cotangent de la variété de drapeaux partiels T^*(G/P). Comme application, nous prouvons un analogue parabolique de l'équivalence de Arkhipov-Bezrukavnikov-Ginzburg. Lorsque la caractéristique de k est supérieure au nombre de Coxeter, nous déduisons un analogue de la conjecture de Finkelberg-Mirkovic graduée pour certains blocs singuliers.

A simple character formula

“…Therefore, the I _,ùorbit in Fl associated with t ς is the inverse image under the projection π : Fl Ñ Gr of the orbit of L ς . Using [ACR,Lemma A.5] we deduce that E IW tς " π ˚pF IW ς qrℓpw f qs. It follows that p N tς " ř zPW f v ℓpzq N tς ¨z , and that for any s P S f we have p N tς ¨Hs " pv `v´1 q ¨pN tς .…”

“…Again, the convolution construction defines a right action of the monoidal category Parity I _ pFl, Kq on the category Parity I _ pGr op , Kq, and there exists a unique isomorphism of right H ext -modules [ACR,Lemma A.5] and the construction of the pcanonical basis in M sph ext , one can check that, for w P f W ext , p M w is the inverse image of rF w s under this isomorphism. 4.2.…”

Co-$t$-structures on derived categories of coherent sheaves and the cohomology of tilting modules

“…The following proposition gives a key property of these objects. For 1 , this statement can be found in [AR,§9.5]; for {1}, see [ACR,Lemma 3.1]. In the special case where I = ∅, the 1 part of this statement was proved in [ARd, MR]; the essential ideas go back to [Be3], which treats the case k = C.…”

A geometric model for blocks of Frobenius kernels

“…(2) We will apply a kind of "highest weight" formalism developed in [BS]. (Note that mod Y (I A u ,XA) (R) is not a highest weight category in the sense considered in, for instance, [ACR,AR1,AR2], because the poset that governs it has no minimal element. )…”