Geometric Satake, Springer correspondence, and small representations II

Pramod N. Achar (1) , Anthony Henderson (2) , Simon Riche (3)

1 LSU Mathematics - Department of Mathematics [Baton Rouge]
2 School of Mathematics and statistics [Sydney]
3 LMBP - Laboratoire de Mathématiques Blaise Pascal

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 $k=\C$ 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.

Mots clés: groupes algébriques réductifs; groupe de Weyl; équivalence de Satake; correspondance de Springer; représentations petites

Domaines: Théorie des représentations [math.RT]

Pramod N. Achar, Anthony Henderson, Simon Riche. Geometric Satake, Springer correspondence, and small representations II. Representation theory, 2015, 19. ⟨hal-00700454⟩