Equivariant perverse sheaves and quasi-hereditary algebras

• Published in: Journal of algebra 2022-02, Vol.591, p.289-341
• Main Author: Joshua, Roy
• Format: Article
• Language: English
• Subjects: Equivariant perverse sheaves; Linear algebraic group actions; Quasi-hereditary algebras
• ISSN: 0021-8693; 1090-266X
• DOI: 10.1016/j.jalgebra.2021.10.027

Let X denote a quasi-projective variety over a field on which a connected linear algebraic group G acts with finitely many orbits. Then, the G-orbits define a stratification of X. We establish several key properties of the category of equivariant perverse sheaves on X, which have locally constant cohomology sheaves on each of the orbits. Under the above assumptions, we show that this category comes close to being a highest weight category in the sense of Cline, Parshall and Scott and defines a quasi-hereditary algebra. We observe that the above hypotheses are satisfied by all toric varieties and by all spherical varieties associated to connected reductive groups over any algebraically closed field. Next we show that the odd dimensional intersection cohomology sheaves vanish on all spherical varieties defined over algebraically closed fields of positive characteristics, extending similar results for spherical varieties defined over the field of complex numbers by Michel Brion and the author in prior work. Assuming that the linear algebraic group G and the action of G on X are defined over a finite field Fq, and where the odd dimensional intersection cohomology sheaves on the orbit closures vanish, we also establish several basic properties of the mixed category of mixed equivariant perverse sheaves so that the associated terms in the weight filtration are finite sums of the shifted equivariant intersection cohomology complexes on the orbit closures.