Geometric Satake, Springer correspondence and small representations
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Geometric Satake, Springer correspondence and small representations Pramod N. Achar · Anthony Henderson
Published online: 5 May 2013 © Springer Basel 2013
Abstract For a simply-connected simple algebraic group G over 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 G L n . Using this variety, we construct a sheaftheoretic 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. Keywords
Affine Grassmannian · Nilpotent orbits · Springer correspondence
Mathematics Subject Classification (2010) 14M15
Primary 17B08, 20G05 · Secondary
1 Introduction Let G be a simply-connected simple algebraic group over C, and Gˇ its Langlands ˇ and let W be dual group. Let T and Tˇ be corresponding maximal tori of G and of G, the Weyl group of either (they are canonically identified). Recall that an irreducible ˇ For representation V of Gˇ is said to be small if no weight of V is twice a root of G. ˇ T such V , the representation of W on the zero weight space V has various special properties, mostly due to Broer [10] and Reeder [27–29].
P. N. Achar (B) Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA e-mail: [email protected] A. Henderson School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australia e-mail: [email protected]
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P. N. Achar, A. Henderson
The aim of this paper is to give a geometric explanation of these properties, using the geometric Satake equivalence (see [16,23,26]) and the Springer correspondence (see [15]). The idea of explaining [10] using geometric Satake was suggested to us by Ginzburg; the idea of explaining [27] using perverse sheaves on the affine Grassmannian was suggested to Reeder by Lusztig, as mentioned in [28]. Let Gr and N denote the affine Grassmannian and the nilpotent cone of G, respectively, and consider the diagram ˇ Rep(G)
Satake ∼
/ PervG(O) (Gr)
(1.1)
Rep(W )
Springer
/ PervG (N )
ˇ
Here, is the functor V → V T ⊗ where denotes the sign representation of W . We will construct a functor which completes diagram (1.1) to a commuting square, after restricting the top line to the subcategories corresponding to small representations. Let Grsm ⊂ Gr be the closed subvariety corresponding to small representations under geometric Satake, and let M ⊂ Grsm be the intersection of Grsm with the ‘opposite Bruhat cell’ Gr− 0 . (See Sect. 2 for detailed definitions.) M is a G-stable dense open subset of Grsm . Our first result is the following. Theorem 1.1 There is an action of Z/2Z on M, commuting with the G-action, and a finite G-equivariant map π : M → N that induces a bijection between M/(Z/2Z) and a certain closed subvariety Nsm of N . The bijection mentioned in Theorem 1.1 is an isomorphism at least
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