Irreducible components of exotic Springer fibres

TL;DR

This paper characterizes the irreducible components of exotic Springer fibres, establishing a bijection with standard bitableaux and introducing an exotic Robinson-Schensted correspondence, simplifying the parametrization compared to classical cases.

Contribution

It describes the irreducible components of exotic Springer fibres and introduces an explicit exotic Robinson-Schensted bijection, simplifying the geometric parametrization in type C.

Findings

Irreducible components are in bijection with standard bitableaux.

Established an explicit exotic Robinson-Schensted bijection.

Identified cases where components are Hirzebruch surfaces.

Abstract

Kato introduced the exotic nilpotent cone to be a substitute for the ordinary nilpotent cone of type C with cleaner properties. Here we describe the irreducible components of exotic Springer fibres (the fibres of the resolution of the exotic nilpotent cone), and prove that they are naturally in bijection with standard bitableaux. As a result, we deduce the existence of an exotic Robinson-Schensted bijection, which is a variant of the type C Robinson-Schensted bijection between pairs of same-shape standard bitableaux and elements of the Weyl group; this bijection is described explicitly in the sequel to this paper. Note that this is in contrast with ordinary type C Springer fibres, where the parametrisation of irreducible components, and the resulting geometric Robinson-Schensted bijection, are more complicated. As an application, we explicitly describe the structure in the special cases where the irreducible components of the exotic Springer fibre have dimension 2, and show that in those cases one obtains Hirzebruch surfaces.