 
Summary: GEOMETRIC SATAKE, SPRINGER CORRESPONDENCE,
AND SMALL REPRESENTATIONS
PRAMOD N. ACHAR AND ANTHONY HENDERSON
Abstract. For a simplyconnected simple algebraic group G over C, we ex
hibit a subvariety of its affine Grassmannian that is closely related to the
nilpotent cone of G, generalizing a wellknown fact about GLn. Using this
variety, we construct a sheaftheoretic functor that, when combined with the
geometric Satake equivalence and the Springer correspondence, leads to a geo
metric explanation for a number of known facts (mostly due to Broer and
Reeder) about small representations of the dual group.
1. Introduction
Let G be a simplyconnected simple algebraic group over C, and G its Langlands
dual group. Let T and T be corresponding maximal tori of G and of G, and let
W be the Weyl group of either (they are canonically identified). Recall that an
irreducible representation V of G is said to be small if no weight of V is twice a
root of G. For such V , the representation of W on the zero weight space V
T
has
various special properties, mostly due to Broer and Reeder [Br1, R1, R2, R3].
The aim of this paper is to give a geometric explanation of these properties, using
