Document Zbl 0819.20044 - zbMATH Open
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Simulating perverse sheaves in modular representation theory. (English) Zbl 0819.20044
Haboush, William J. (ed.) et al., Algebraic groups and their generalizations: classical methods. Summer Research Institute on algebraic groups and their generalizations, July 6-26, 1991, Pennsylvania State University, University Park, PA, USA. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 56, Pt. 1, 63-104 (1994).
This paper is closely connected with the authors’ paper [in Tôhoku Math. J., II. Ser. 45, 511-534 (1994; Zbl 0801.20013)]. Both of them are devoted to the investigation of the questions: which properties of perverse sheaves can be proved in the framework of a general highest weight category? Which ingredients are necessary to reproduce the geometric arguments of MacPherson used in the proof of the Kazhdan- Lusztig conjecture for the composition factor multiplicities of Verma modules?
The present paper contains, together with a review of the main results of the paper cited above, the details and issues more strictly concerned with the extent to which general highest weight categories imitate perverse sheaf categories. An algebraic version of the Deligne-MacPherson characterization of perverse sheaves is given and a \(t\)-structure and algebraic cohomology groups capturing the ordinary cohomology groups of perverse sheaves, as well as the “parity theorem” (a weak replacement for Gabber’s purity theorem) are discussed.
For the entire collection see [Zbl 0793.00018].