Reductive groups, the loop Grassmannian, and the Springer resolution

Abstract

In this paper we prove equivalences of categories relating the derived category of a block of the category of representations of a connected reductive algebraic group over an algebraically closed field of characteristic p bigger than the Coxeter number and a derived category of equivariant coherent sheaves on the Springer resolution (or a parabolic counterpart). In the case of the principal block, combined with previous results, this provides a modular version of celebrated constructions due to Arkhipov-Bezrukavnikov-Ginzburg for Lusztig's quantum groups at a root of unity. As an application, we prove a "graded version" of a conjecture of Finkelberg-Mirković describing the principal block in terms of mixed perverse sheaves on the dual affine Grassmannian, and deduce a new proof of Lusztig's conjecture in large characteristic. Part 2. Formality theorems 34 5. Formality for P I,1 -modules 34 6.Ṗ J -equivariant formality 41 7. Compatibility with induction 49 Part 3. Induction theorems 58 8. Translation functors 58 9. Cotangent bundles of partial flag varieties 75 10. The induction theorem 92 11. The graded Finkelberg-Mirković conjecture 97 Index of notation 109 References 110 P.A.

An Iwahori-Whittaker model for the Satake category

“…Let k be an algebraic closure of k, and assume that the characteristic of k is strictly bigger than the Coxeter number of G. Then the formula of Proposition 4.18 is related to Donkin's tensor product theorem for tilting modules of the Langlands dual k-group G ∨ k as follows. In [RW,AR2,AMRW] the authors construct a "degrading functor" η : Parity IW (Fl • , k) → Tilt prin (G ∨ k ), where Fl • is the connected component of the base point in Fl, Parity IW (Fl • , k) is the category of (I + u , χ * I + (L k ψ ))-equivariant parity complexes on Fl • , and Tilt prin (G ∨ k ) denotes the category of tilting objects in the (non-extended) principal block of the category of finite-dimensional G ∨ k -modules. We expect that Donkin's tensor product theorem (see [Ja,§E.9]) can be explained geometrically by an isomorphism of complexes involving the functor Z .…”

Algebraic representations and constructible sheaves

“…At present we know much more about the category of algebraic representations than of kGpF q q (e.g. compare the induction theorems of [ABG04,HKS16,AR16c] with the solution of Broué's conjecture for SL 2 pF q q [Chu01, Oku00]).…”

“…Another grading on Rep 0 is constructed in [AR16c]. These two gradings should be related by Koszul duality.…”

Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

“…This paper draws inspiration from [AR1], which (in the setting of parity complexes on flag varieties) introduced the notions of "mixed derived category" and "mixed perverse sheaves." These notions have since found important applications in modular representation theory; see in particular [AR1,ARd2,MaR,AR2,AMRW2].…”

“…In[AR2], for simplicity this claim is stated only in the case k is a field. But the same arguments apply in the present generality; see e.g.…”