On a reducedness conjecture for spherical Schubert varieties and slices in the affine Grassmannian

TL;DR

This paper investigates the scheme-theoretic validity of a modular description of spherical Schubert varieties in the affine Grassmannian and explores its connection to a reducedness conjecture for slices, providing partial proofs and insights.

Contribution

It proves the modular description is scheme-theoretically correct in many cases and links it to the reducedness conjecture for slices in the affine Grassmannian.

Findings

Modular description is set-theoretically correct.

Proved scheme-theoretic correctness in several cases.

Connected the description to the reducedness conjecture.

Abstract

We study spherical Schubert varieties in the affine Grassmannian. These Schubert varieties have a natural conjectural modular description due to Finkelberg-Mirkovi\'c. This modular description is easily seen to be set-theoretically correct, but it is not obviously scheme-theoretically correct. We prove that this modular description is correct in many cases. We also link this modular description to the reducedness conjecture from Kamnitzer-Webster-Weekes-Yacobi for tranverse slices in the affine Grassmannian.