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hal-00700454, version 1

Geometric Satake, Springer correspondence, and small representations II

Pramod N. Achar 1, Anthony Henderson 2, Simon Riche (Author to contact preferably) 3

(2012-05-23)

  • 1:  Department of Mathematics [Baton Rouge] (LSU Mathematics)
  • https://www.math.lsu.edu/
    Louisiana State University Department of Mathematics 303 Lockett Hall * Baton Rouge, LA 70803-4918 United States
  • 2:  School of Mathematics and statistics [Sydney]
  • http://www.maths.usyd.edu.au/
    The University of Sydney School of Mathematics and Statistics F07 University of Sydney NSW 2006 Australia Australia
  • 3:  Laboratoire de Mathématiques
  • http://math.univ-bpclermont.fr/index.html
    CNRS : UMR6620 – Université Blaise Pascal - Clermont-Ferrand II 24 avenue des Landais 63177 AUBIERE CEDEX France

Bibliographic reference

  • Type of document: Documents without publication reference (Preprint)
  • Subject: Mathematics/Representation Theory
  • Title: 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 $k=\C$ 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.
  • Fulltext language: English
  • Production date: 2012-05-23
  • Keyword(s): groupes algébriques réductifs – groupe de Weyl – équivalence de Satake – correspondance de Springer – représentations petites
  • Comment: 83 pages

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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