Study of some orthosymplectic Springer fibers

TL;DR

This paper investigates the structure of Springer fibers associated with the odd nilcone of a specific Lie superalgebra, revealing their connectedness, disconnection, and dimensional properties through a novel decomposition approach.

Contribution

It introduces a decomposition of Springer fibers for the orthosymplectic Lie superalgebra, demonstrating their potential disconnection and non-equidimensionality, contrasting with classical cases.

Findings

Most fibers are connected.

Fibers can be disconnected.

Fibers may be non-equidimensional.

Abstract

We decompose the fibers of the Springer resolution for the odd nilcone of the Lie superalgebra $\osp(2n+1,2n)$ into locally closed subsets. We use this decomposition to prove that almost all fibers are connected. However, in contrast with the classical Springer fibers, we prove that the fibers can be disconnected and non equidimensional.