## University of Sydney Algebra Seminar

# Anthony Henderson (Sydney)

## Friday 17 October, 12:05-12:55pm, Place: 373

### Modular generalized Springer correspondence

### The motivation behind this project is the desire for a theory of modular character sheaves, which would provide a geometric interpretation of the decomposition matrix of a finite group of Lie type in non-defining characteristic. As a first step, we have established a modular generalized Springer correspondence for any connected reductive complex group.

This has a similar form to Lusztig's generalized Springer correspondence: one has a partition of the simple equivariant perverse sheaves on the nilpotent cone into induction series parametrized by cuspidal perverse sheaves for various Levi subgroups, and the elements of each induction series are in bijection with irreducible representations of the relative Weyl group of the Levi. Our setting involves representations and perverse sheaves over a field of positive characteristic, which creates many difficulties and new features. Compared to Lusztig's situation, there are more cuspidal perverse sheaves; in particular, the Levi subgroups that support them do not have to be self-opposed, and the relative Weyl groups do not have to be Coxeter groups. In type A, the classification of cuspidals and the explicit determination of each series is complete, and is appropriately reminiscent of known results on unipotent Brauer characters for the finite general linear group. In other classical types, the classification of cuspidals is complete, but we have determined the induction series only in characteristic 2. In the exceptional types, several indeterminacies remain.

This is joint work with Pramod Achar, Daniel Juteau and Simon Riche.