Geometric Satake, Springer correspondence, and small representations

TL;DR

This paper links affine Grassmannian subvarieties with the nilpotent cone to provide a geometric understanding of small representations of the dual group, integrating geometric Satake and Springer correspondence.

Contribution

It introduces a new subvariety of the affine Grassmannian related to the nilpotent cone and constructs a sheaf-theoretic functor that explains properties of small representations.

Findings

Establishment of a subvariety related to the nilpotent cone

Construction of a sheaf-theoretic functor connecting geometric Satake and Springer

Geometric explanation for properties of small representations

Abstract

For a simply-connected simple algebraic group $G$ over $\C$, we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of $G$, generalizing a well-known fact about $GL_n$. Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.