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Representation Theory

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Geometric Satake, Springer correspondence, and small representations II

Authors: Pramod N. Achar, Anthony Henderson and Simon Riche
Journal: Represent. Theory 19 (2015), 94-166
MSC (2010): Primary 17B08, 20G05; Secondary 14M15
Published electronically: May 18, 2015
MathSciNet review: 3347990
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Abstract: 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

Anthony Henderson
Affiliation: School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Simon Riche
Affiliation: Université Blaise Pascal et CNRS, Laboratoire de Mathématiques (UMR 6620), Campus universitaire des Cézeaux, F-63177 Aubière Cedex, France

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)
Article copyright: © Copyright 2015 American Mathematical Society