Toric varieties and a generalization of the Springer resolution

TL;DR

This paper introduces a new construction of an analogue to the Springer resolution for the universal cover of a nilpotent orbit using toric varieties, extending the classical theory and providing new proofs of existing results.

Contribution

It develops a novel Springer resolution analogue for the universal cover of a nilpotent orbit employing toric variety theory, generalizing previous constructions.

Findings

Provides a new proof of Broer and Graham's results

Constructs Springer resolution analogue for universal covers

Extends the construction to arbitrary nilpotent orbit covers

Abstract

The Springer resolution of the nilpotent cone of a semisimple Lie algebra has played an important role in representation theory. The nilpotent cone is equal to Spec R, where R is the ring of regular functions on the nilpotent cone. This paper constructs and studies an analogue of the Springer resolution for the variety Spec S, where S is the ring of regular functions on the universal cover of the regular nilpotent orbit. The construction makes use of the theory of toric varieties. Using this construction, we provide new proofs of results of Broer and of Graham about Spec S. Finally, we show that the construction can be adapted to covers of an arbitrary nilpotent orbit.