Geometric Satake, Springer correspondence, and small representations

Pramod N. Achar and Anthony Henderson

Abstract

For a simply-connected simple algebraic group \(G\) over \(\mathbb{C}\), 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.

Keywords: algebraic group, nilpotent orbits, representations.

AMS Subject Classification: Primary 17B08, 20G05; Secondary 14M15.