Let G be a connected reductive algebraic group over C. Let Λ + G be the monoid of dominant weights of G. We construct the integrable crystals B G (λ), λ ∈ Λ + G , using the geometry of generalized transversal slices in the affine Grassmannian of the Langlands dual group. We construct the tensor product maps p λ 1 ,λ 2 :in terms of multiplication of generalized transversal slices. Let L ⊂ G be a Levi subgroup of G. We describe the restriction to Levi Res G L : Rep(G) → Rep(L) in terms of the hyperbolic localization functors for the generalized transversal slices.
Let G be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian Gr G . We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with U (n ∨ ) (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra g ∨ ). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.
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