Cotangent Bundle to the Flag Variety - II

TL;DR

This paper constructs a natural compactification of the cotangent bundle to flag varieties using affine Schubert varieties, providing a new geometric perspective and recovering the Springer resolution for nilpotent orbit closures.

Contribution

It introduces a novel compactification of cotangent bundles to flag varieties via affine Schubert varieties within an infinite-dimensional Kac-Moody setting.

Findings

Constructs a $SL_n(C)$-stable closed subvariety in an affine Schubert variety.

Provides a geometric realization of the cotangent bundle compactification.

Recovers the Springer resolution for nilpotent orbit closures.

Abstract

Let $P$ be a parabolic subgroup in $SL_n(\mathbb C)$. We show that there is a $SL_n(\mathbb C)$-stable closed subvariety of an affine Schubert variety in an infinite dimensional partial Flag variety (associated to the Kac-Moody group ${\widehat{SL_n}(\mathbb C)}$) which is a natural compactification of the cotangent bundle to $SL_n(\mathbb C)/P$. As a consequence, we recover the Springer resolution for any orbit closure inside the variety of nilpotent matrices.