Decomposition numbers for perverse sheaves

TL;DR

This paper develops foundational methods for calculating decomposition numbers of perverse sheaves, providing concrete computations for specific singularities and orbit closures with applications in modular representation theory.

Contribution

It introduces new methods to compute decomposition numbers of perverse sheaves and applies them to Kleinian surface singularities and minimal orbit closures.

Findings

Computed decomposition numbers for Kleinian surface singularities

Determined decomposition numbers for minimal orbit closures in Lie algebras

Established foundational techniques for broader applications

Abstract

The purpose of this article is to set foundations for decomposition numbers of perverse sheaves, to give some methods to calculate them in simple cases, and to compute them concretely in two situations: for a simple (Kleinian) surface singularity, and for the closure of the minimal non-trivial orbit in a simple Lie algebra. This work has applications to modular representation theory, for Weyl groups using the nilpotent cone of the corresponding semisimple Lie algebra, and for reductive algebraic group schemes using the affine Grassmannian of the Langlands dual group.