Geometric Satake, Springer correspondence, and small representations II

Abstract

For a split reductive group scheme $G$ over a commutative ring $k$ with Weyl group $W$, there is an important functor $Rep(G,k)\to Rep(W,k)$ defined by taking the zero weight space. We prove that the restriction of this functor to the subcategory of small representations has an alternative geometric description, in terms of the affine Grassmannian and the nilpotent cone of the Langlands dual group to $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $k=\mathbb{C}$ case proved by the first two authors, and also provides a better explanation than in that earlier paper, since the current proof is uniform across all types.

Version 4: minor revisions; 73 pages

References in corpus (6)

• Higher Topos Theory
• Enhanced six operations and base change theorem for higher Artin stacks
• Quiver varieties and Beilinson-Drinfeld Grassmannians of type A
• Perverse coherent sheaves and the geometry of special pieces in the unipotent variety
• Correspondance de Springer modulaire et matrices de décomposition
• A geometric Schur functor

Cited by in corpus (3)

• Modular Springer correspondence, decomposition matrices and basic sets
• A geometric Schur functor
• Modular generalized Springer correspondence I: the general linear group