Modular perverse sheaves on flag varieties. I: Tilting and parity sheaves. (Q2811181)

Let \(G\) be a complex connected reductive group with a Borel subgroup \(B\), and let \(\check{G}\) is the Langlands dual group of \(G\) with a Borel subgorup \(\check{B}\). Let \(\mathcal{B}\) be the flag variety of \(G\), and let \(\check{\mathcal{B}}\) be the flag variety of \(\check{G}\). Let \(\mathbb{K}\) be a field of characteristic of characteristic \(\ell\).NEWLINENEWLINEThe main result of this paper is the construction of an equivalence between the category \(\text{Parity}_{\check{B}}(\check{\mathcal{B}}, \mathbb{K})\) of \(\check{B}\)-equivariant parity complexes with coefficients in \(\mathbb{K}\) on \(\check{\mathcal{B}}\) and the category \(\text{Tilt}_{B}(\mathcal{B}, \mathbb{K})\) of \(B\)-equivariant tilting perverse sheaves on \(\mathcal{B}\). One can view that the category \(\text{Parity}_{\check{B}}(\check{\mathcal{B}}, \mathbb{K})\) realizes a graded version of the category \(\text{Tilt}_{B}(\mathcal{B}, \mathbb{K})\).NEWLINENEWLINEAs an application of the main construction, the authors prove that when the characteristic \(\ell\) of \(\mathbb{K}\) is greater than the Coxeter nubmer of \(G\), then the category \(P_B(\mathcal{B}, \mathbb{K})\) of \(B\)-equivariant perverse sheaves on \(\mathcal{B}\) is equivalent, as a highest weight category , to Soergel's modular categorh \(\mathcal{O}\) which is associated to the conected reductibe group \(G_{\mathbb{K}}\) over \(\mathbb{K}\) which has the same root datum as \(G\).NEWLINENEWLINECombining with the Koszul self-duality which sends tilting perverse sheaves to simple perverse sheaves, and the main construction of this paper, the authors of this paper prove that the decomposition numbers in Soergel's modular category \(\mathcal{O}\) associated to \(G_{\mathbb{K}}\) can be computed by the stalks of parity sheaves on the Langlands dual group \(\check{G}\). This is a modular analogue of Kahdan-Lusztig conjecture, which asserts that the multiplicity of simple modules in category \(\mathcal{O}\) in Verma modules can be computed by the Kazhdan-Lusztig polynomials, which can be exactly obtaiend by the stalks of simple perverse sheaves on flag variety. The simple perverse sheaves on flag variety gives rise to Kazhdan-Lusztig basis in the Hecke algebra, and parallelly the parity sheaves gives rise to \(\ell\)-canonical basis.