The category O of slices in the affine Grassmannian


The category O of slices in the affine Grassmannian

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The affine Grassmannian GrG is an important algebraic (ind-)variety in geometric representation theory associated to a reductive group G. The slices in GrG are naturally occurring subvarieties which, by the geometric Satake correspondence of Mirkovic and Vilonen, geometrise weight spaces of irreducible representations of G^L, the Langlands dual group. They carry a natural Poisson structure, and under symplectic duality (due to Braden, Licata, Proudfoot, and Webster) they are dual to another class of important varieties called Nakajima quiver varieties. The essential feature of this duality is formulated as a Koszul duality between categories associated to these varieties called categories O (these categories generalise the usual BGG category O of g=Lie(G)-modules).

I will explain these ideas in a basic example, and use this to motivate the study of the category O associated to the slices in the affine Grassmannian. The main result I want to explain is a combinatorial description of the set of simple objects in this category, which turns out is governed by a finite dimensional representation of g^L. We conjectured this description in 2014, and recently proved it by relating the category to Webster's tensor product algebras. I will try to explain the basic ideas of this proof.

This work is joint with various subsets of {J. Kamnitzer, P. Tingley, B. Webster, A. Weekes}.


  • Dr Oded Yacobi
    Dr Oded Yacobi, University of Sydney