Relation between two geometrically defined bases in representations of $G{L}_n$

Abstract

Let $V$ be an irreducible representation of group $G{L}_n(\mathbb{C})$, which appears as a submodule in $({\mathbb{C}}^n{)}^{\otimes d}$, where ${\mathbb{C}}^n$ is the tautological $n$-dimensional representation of $G{L}_n$, and $d$ is a non-negative integer. On the one hand, following refs [Gi] and [BG] one can produce a basis in $V$ using irreducible components of Sringer fibers over a nilpotent matrix in ${\mathfrak{gl}}_d$, whose Jordan blocks correspond to the highest weight of $V$. On the other hand, one can produce a basis in $V$ by Mirković-Vilonen cycles, a construction that works for an arbitrary reductive group $G$. In this note we prove that the resulting to bases coincide.