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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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Geometric Satake, Springer correspondence, and small representations II
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by Pramod N. Achar, Anthony Henderson and Simon Riche PDF
Represent. Theory 19 (2015), 94-166 Request permission


For a split reductive group scheme $\check G$ over a commutative ring $\Bbbk$ with Weyl group $W$, there is an important functor ${\mathsf {Rep}}(\check G,\Bbbk )\to {\mathsf {Rep}}(W,\Bbbk )$ 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 $G$. The translation from representation theory to geometry is via the Satake equivalence and the Springer correspondence. This generalizes the result for the $\Bbbk =\mathbb {C}$ case proved by the first two authors, and also provides a better explanation than in the earlier paper, since the current proof is uniform across all types.
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Additional Information
  • Pramod N. Achar
  • Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803
  • MR Author ID: 701892
  • Email:
  • Anthony Henderson
  • Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
  • MR Author ID: 687061
  • ORCID: 0000-0002-3965-7259
  • Email:
  • Simon Riche
  • Affiliation: Université Blaise Pascal et CNRS, Laboratoire de Mathématiques (UMR 6620), Campus universitaire des Cézeaux, F-63177 Aubière Cedex, France
  • MR Author ID: 834430
  • Email:
  • Received by editor(s): January 31, 2014
  • Published electronically: May 18, 2015
  • Additional Notes: The first author was supported by NSF Grant No. DMS-1001594. The second author was supported by ARC Future Fellowship Grant No. FT110100504. The third author was supported by ANR Grants No. ANR-09-JCJC-0102-01 and No. ANR-2010-BLAN-110-02.

  • Dedicated: In memoriam T. A. Springer (1926–2011)
  • © Copyright 2015 American Mathematical Society
  • Journal: Represent. Theory 19 (2015), 94-166
  • MSC (2010): Primary 17B08, 20G05; Secondary 14M15
  • DOI:
  • MathSciNet review: 3347990