On modular categories O for quantized symplectic resolutions

2017/12/20 by Losev, Ivan · 1 citation
#16E35 #16G99 #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1712.07726

Abstract

In this paper we study highest weight and standardly stratified structures on modular analogs of categories O over quantizations of symplectic resolutions and show how to recover the usual categories O (reduced mod p≫ 0) from our modular categories. More precisely, we consider a conical symplectic resolution that is defined over a finite localization of ℤ and is equipped with a Hamiltonian action of a torus T that has finitely many fixed points. We consider algebras A_λ of global sections of a quantization in characterstic p≫ 0, where λ is a parameter. Then we consider a category O_λ consisting of all finite dimensional T-equivariant A_λ-modules. We show that for λ lying in a \it p-alcove ^p A, the category O_λ is highest weight (in some generalized sense). Moreover, we show that every face of ^p A that survives in ^p A/p when p→ ∞ defines a standardly stratified structure on O_λ. We identify the associated graded categories for these standardly stratified structures with reductions mod p of the usual categories O in characteristic 0. Applications of our construction include computations of wall-crossing bijections in characteristic p and the existence of gradings on categories O in characteristic 0.

Cited by

• Cactus flower spaces and monodromy of Bethe vectors

Related

• Wall-crossing functors for quantized symplectic resolutions: perversity and partial Ringel dualities
• The canonical global quantization of symplectic varieties in characteristic p
• Stratifications of abelian categories
• Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties
• Category O and asymptotic characters