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 Geometric Satake, Springer correspondence, and small representations II
 Pramod N. Achar 1, Anthony Henderson 2, Simon Riche ( ) 3
 (2012-05-23)
 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.
 1: Department of Mathematics [Baton Rouge] (LSU Mathematics) Louisiana State University 2: School of Mathematics and statistics [Sydney] The University of Sydney 3: Laboratoire de Mathématiques CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II
 Subject : Mathematics/Representation Theory
 Keyword(s): groupes algébriques réductifs – groupe de Weyl – équivalence de Satake – correspondance de Springer – représentations petites
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 hal-00700454, version 1 http://hal.archives-ouvertes.fr/hal-00700454 oai:hal.archives-ouvertes.fr:hal-00700454 From: Simon Riche <> Submitted on: Wednesday, 23 May 2012 10:26:50 Updated on: Wednesday, 23 May 2012 11:29:04