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Geometric Satake, Springer correspondence, and small representations. II. (English) Zbl 1407.17008

The authors deal with the functor \(\mathrm{Rep}(\tilde{G}, k) \mapsto\mathrm{Rep}(\mathcal{W}, k), \) defined by taking the zero weight space, where \(\tilde{G}\) is a split reductive group scheme over a commutative ring \(k\) with Weyl group \(\mathcal{W}.\) They 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 \(\tilde{G}.\) Indeed, they improve and generalize previous results by Satake and Springer, who proved the \(k = \mathbb{C}\) case. They force the arguments which they use in the paper to be \(2\)-categorical, that is to say, that showing that a diagram of functors commutes it is not enough to show the existence of an isomorphism of functors, since that it must keep track of what the isomorphism is. In this way, the main arguments of this paper could be carried out in the framework of \(\infty\)-categories developed by Boardman-Vogt, Joyal and Lurie, among others. Obviously, working with \(\infty\)-categories offers certain advantages with respect to do it with \(2\)-categories but the prefer these last ones because of they are more accessible to begin being studied.
For Part I, see the first two authors [Sel. Math., New Ser. 19, No. 4, 949–986 (2013; Zbl 1319.17003)].

MSC:

17B08 Coadjoint orbits; nilpotent varieties
20G05 Representation theory for linear algebraic groups
14L35 Classical groups (algebro-geometric aspects)

Citations:

Zbl 1319.17003
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