Springer basic sets and modular Springer correspondence for classical types

TL;DR

This paper introduces basic set data for finite groups, demonstrates that Springer correspondence provides such data for Weyl groups, and explicitly determines the modular Springer correspondence for classical types in odd characteristic.

Contribution

It defines basic set data incorporating an order on irreducible characters, and uses this to explicitly compute the modular Springer correspondence for classical types.

Findings

Springer correspondence provides basic set data for Weyl groups.

Explicit determination of modular Springer correspondence for classical types.

Comparison of bipartition orders and a new proof method using Springer fiber dimensions.

Abstract

We define the notion of basic set data for finite groups (building on the notion of basic set, but including an order on the irreducible characters as part of the structure), and we prove that the Springer correspondence provides basic set data for Weyl groups. Then we use this to determine explicitly the modular Springer correspondence for classical types (defined over a base field of odd characteristic $p$, and with coefficients in a field of odd characteristic $\ell\neq p$): the modular case is obtained as a restriction of the ordinary case to a basic set. In order to do so, we compare the order on bipartitions introduced by Dipper and James with the order induced by the Springer correspondence. We also provide a quicker proof, by sorting characters according to the dimension of the corresponding Springer fiber, an invariant which is directly computable from symbols.