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Geometric Satake, Springer correspondence and small representations

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Abstract

For a simply-connected simple algebraic group \(G\) over \(\mathbb C \), we exhibit a subvariety of its affine Grassmannian that is closely related to the nilpotent cone of \(G\), generalizing a well-known fact about \(GL_n\). Using this variety, we construct a sheaf-theoretic functor that, when combined with the geometric Satake equivalence and the Springer correspondence, leads to a geometric explanation for a number of known facts (mostly due to Broer and Reeder) about small representations of the dual group.

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Acknowledgments

This paper developed from discussions with V. Ginzburg and S. Riche, to whom the authors are much indebted. In particular, V. Ginzburg posed the problem of finding a geometric interpretation of Broer’s covariant theorem in the context of geometric Satake. Much of the work was carried out during a visit by P.A. to the University of Sydney in May–June 2011, supported by ARC Grant No. DP0985184. P.A. also received support from NSF Grant No. DMS-1001594.

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Pramod N. Achar

  2. School of Mathematics and Statistics, University of Sydney, Sydney, NSW, 2006, Australia

    Anthony Henderson

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  1. Pramod N. Achar
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  2. Anthony Henderson
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Correspondence to Pramod N. Achar.

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Achar, P.N., Henderson, A. Geometric Satake, Springer correspondence and small representations. Sel. Math. New Ser. 19, 949–986 (2013). https://doi.org/10.1007/s00029-013-0125-7

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