T-Surfaces in the Affine Grassmannian

Abstract

In this thesis we examine singularities of surfaces and affine Schubert varieties in the affine Grassmannian $\mathcal{G}/\mathcal{P}$ of type $A^{(1)}$, by considering the action of a particular torus $\widehat{T}$ on $\mathcal{G}/\mathcal{P}$. Let $\Sigma$ be an irreducible $\widehat{T}$-stable surface in $\mathcal{G}/\mathcal{P}$ and let $u$ be an attractive $\widehat{T}$-fixed point with $\widehat{T}$-stable affine neighborhood $\Sigma_u$. We give a description of the $\widehat{T}$-weights of the tangent space $T_u(\Sigma)$ of $\Sigma$ at $u$, give some conditions under which $\Sigma$ is nonsingular at $u$, and provide some explicit criteria for $\Sigma_u$ to be normal in terms of the weights of $T_u(\Sigma)$. We will also prove a conjecture regarding the singular locus of an affine Schubert variety in $\mathcal{G}/\mathcal{P}$.