## Constructible sheaves on nilpotent cones in rather good characteristic

### Pramod N. Achar, Anthony Henderson, Daniel Juteau, and Simon Riche

#### Abstract

We study some aspects of modular generalized Springer theory for a complex reductive group $$G$$ with coefficients in a field $$\mathbb{k}$$ under the assumption that the characteristic $$\ell$$ of $$\mathbb{k}$$ is rather good for $$G$$, i.e., $$\ell$$ is good and does not divide the order of the component group of the centre of $$G$$. We prove a comparison theorem relating the characteristic-$$\ell$$ generalized Springer correspondence to the characteristic-$$0$$ version. We also consider Mautner's characteristic-$$\ell$$ `cleanness conjecture'; we prove it in some cases; and we deduce several consequences, including a classification of supercuspidal sheaves and an orthogonal decomposition of the equivariant derived category of the nilpotent cone.

Keywords: Nilpotent cone; Springer theory.

: Primary 17B08, 20G05.

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 Tuesday, August 4, 2015