Geometric Satake, Springer correspondence, and small representations II

Achar, Pramod; Henderson, Anthony; Riche, Simon

doi:10.1090/ert/465

Geometric Satake, Springer correspondence, and small representations II
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by Pramod N. Achar, Anthony Henderson and Simon Riche

Represent. Theory 19 (2015), 94-166

DOI: https://doi.org/10.1090/ert/465

Published electronically: May 18, 2015

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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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