Selecta Mathematica

, Volume 19, Issue 4, pp 949–986 | Cite as

Geometric Satake, Springer correspondence and small representations



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.


Affine Grassmannian Nilpotent orbits Springer correspondence 

Mathematics Subject Classification (2010)

Primary 17B08 20G05 Secondary 14M15 



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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© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsLouisiana State UniversityBaton RougeUSA
  2. 2.School of Mathematics and StatisticsUniversity of SydneySydneyAustralia

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