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Tangent spaces of spherical Schubert varieties and a counterexample to the reducedness conjecture

Marc Besson, Jiuzu Hong, Huanhuan Yu

Abstract

Given a simply-connected simple algebraic group $G$, we determine the tangent space of the Finkelberg-Mirković Schubert scheme in the affine Grassmannian of $G$ associated to the quasi-minuscule coweight. As a consequence, we exhibit a non-reduced Finkelberg-Mirković Schubert scheme when $G$ is of type $E_8$.

Tangent spaces of spherical Schubert varieties and a counterexample to the reducedness conjecture

Abstract

Given a simply-connected simple algebraic group , we determine the tangent space of the Finkelberg-Mirković Schubert scheme in the affine Grassmannian of associated to the quasi-minuscule coweight. As a consequence, we exhibit a non-reduced Finkelberg-Mirković Schubert scheme when is of type .
Paper Structure (4 sections, 7 theorems, 46 equations)

This paper contains 4 sections, 7 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Tangent spaces of affine Schubert Varieties: Polo's criterion
  4. Tangent spaces of FM Schubert schemes

Key Result

Theorem 3.1

For $\lambda\in X_*(T)^+$, we have an isomorphism as representations of $G\times \mathbb{G}_m$.

Theorems & Definitions (16)

  • Theorem 3.1
  • Proposition 3.2
  • proof
  • proof
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • ...and 6 more