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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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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DOI: https://doi.org/10.1007/s00029-013-0125-7