Modular perverse sheaves on flag varieties I: tilting and parity sheaves

Abstract

Abstract. In this paper we prove that the category of parity complexes on the flag variety of a complex connected reductive group G is a "graded version" of the category of tilting perverse sheaves on the flag variety of the dual group G, for any field of coefficients whose characteristic is good for G. We derive some consequences on Soergel's modular category O, and on multiplicities and decomposition numbers in the category of perverse sheaves.1. Introduction 1.1. This paper is the first in a series devoted to investigating the structure of the category of Bruhat-constructible perverse sheaves on the flag variety of a complex connected reductive algebraic group, with coefficients in a field of positive characteristic. In this part, adapting some constructions of Bezrukavnikov-Yun [13] in the characteristic 0 setting, we show that in good characteristic, the category of parity sheaves on the flag variety of a reductive group is a "graded version" of the category of tilting perverse sheaves on the flag variety of the Langlands dual group. We also derive a number of interesting consequences of this result, in particular on the computation of multiplicities of simple perverse sheaves in standard perverse sheaves, on Soergel's "modular category O," and on decomposition numbers.1.2. Some notation. Let G be a complex connected reductive algebraic group, and let T ⊂ B ⊂ G be a maximal torus and a Borel subgroup. The choice of B determines a choice of positive roots of (G, T ), namely those appearing in Lie(B). Consider also the Langlands dual dataŤ ⊂B ⊂Ǧ. That is,Ǧ is a complex connected reductive group, and we are given an isomorphism X * (T ) ∼ = X * (Ť ) which identifies the roots of G with the coroots ofǦ (and the positive roots determined by B with the positive coroots determined byB).We are interested in the varieties B := G/B andB :=Ǧ/B, in the derived categories D of sheaves of k-vector spaces on these varieties, constructible with respect to the stratification by B-orbits, resp.B-orbits (where k is field), and in their abelian subcategories P (B) (B, k), resp. P (B) (B, k) of perverse sheaves (for the middle perversity). The category P (B) (B, k) is highest weight, with simple objects {IC w , w ∈ W }, standard objects {∆ w , w ∈ W }, costandard objects {∇ w , w ∈ W }, indecomposable projective objects {P w , w ∈ W } and indecomposable tilting objects {T w , w ∈ W } naturally parametrized by the Weyl group W of (G, T ). Similar remarks apply of course to P (B) (B, k), and we denote the corresponding objects byǏC w ,∆ w ,∇ w ,P w ,Ť w . (Note that the Weyl group of (Ǧ,Ť ) is canonically identified with W .)1.3. The case k = C. These categories have been extensively studied in the case k = C: see in particular [10,9,13]. To state some of their properties we need some notation. We will denote by IC (B) (B, C) the additive category of semisimple objects in D (B, C) (i.e. the full subcategory whose objects are direct sums of shifts of simple perverse sheaves). If A is an abelian category, we will denote by Proj-A ...

Koszul duality for Kac–Moody groups and characters of tilting modules

“…The same arguments as in [AR2,§5.11], using the results of [AR3,§4.4], show that P is the projective cover of the simple object T 1 in the abelian category Perv(U G /B, k). In particular, using (2.6) and (2.8) we deduce that we have…”

“…the triangulated category of (U J U − J , χ * J L ψ )-equivariant complexes on the indvariety G /B (see [AR2,Definition A.1]). Note that if w ∈ W , then X Wh,J w supports a (U J U − J , χ * J L ψ )-equivariant local system if and only if w ∈ J W .…”

Exotic tilting sheaves, parity sheaves on affine Grassmannians, and the Mirković–Vilonen conjecture

“…We learnt how useful this observation can be in [BY]. It also plays an important role in [AR1]. In practice, this lemma can be used when we are interested in modules over a k-algebra B which is "complicated" or not well understood, but for which we have a "simplified model" A which is "isomorphic to B up to torsion", i.e.…”

“…One of the crucial ideas in our proof, which 2 Our "Soergel bimodules" are in fact not bimodules over any ring, but rather modules over a certain algebra built out of two copies of a polynomial algebra. We use this terminology since these objects play the role which is usually played by actual Soergel bimodules. we learnt in papers of Soergel [S1, S3] and was used also in [AR1], is to compute Hom-spaces between Soergel bimodules from the analogue for parity sheaves. We use this computation to prove the "coherent side"; see the proof of Theorem 5.14 for more details.…”

Modular perverse sheaves on flag varieties III: Positivity conditions

“…These are certain representations of an algebraic group, obtained by modular reduction of irreducible quantum group representations. It is likely that under the equivalence of [AR2,Theorem 2.4], reduced standard modules correspond to objects of the form F ⊗ L IC w (O). With this in mind, condition (2a) should be compared to [CPS,Conjecture 6.5], and condition (2c) to [CPS,Conjecture 6.2].…”