Generating varieties for affine Grassmannians

TL;DR

This paper investigates the topological group structure of affine Grassmannians, identifying specific Schubert varieties that generate both the homology ring and the entire affine Grassmannian.

Contribution

It introduces a canonical family of generating Schubert varieties defined by negative coroots, establishing their role in generating the homology and the affine Grassmannian.

Findings

Identifies Schubert varieties that generate the homology ring.

Shows these varieties generate the affine Grassmannian itself.

Provides a canonical family of generators related to the highest root.

Abstract

We study the topological group structure (coming from loop multiplication) on an affine Grassmannian. In particular, we study finite-dimensional subvarieties that generate the homology ring. We show that there is a canonical family of generating Schubert varieties, namely those defined by the negative of the coroot associated to the highest root. These not only generate the homology, but generate the affine Grassmannian itself in a precise sense.