Total positivity in Springer fibres

TL;DR

This paper investigates the structure of Springer fibres at unipotent elements within the totally positive part of a complex reductive group, revealing a natural cell decomposition aligned with the positive flag manifold's structure.

Contribution

It establishes a natural cell decomposition of the intersection of Springer fibres at unipotent elements with the totally positive flag manifold, connecting it to Rietsch's decomposition.

Findings

The intersection has a natural cell decomposition.

This decomposition aligns with Rietsch's cell decomposition of the positive flag manifold.

The work links Springer fibres with total positivity structures.

Abstract

Let u be a unipotent element in the totally positive part of a complex reductive group. We consider the intersection of the Springer fibre at u with the totally positive part of the flag manifold. We show that this intersection has a natural cell decomposition which is part of the cell decomposition (Rietsch) of the totally positive flag manifold.