Integrable crystals and restriction to Levi via generalized slices in the affine Grassmannian

2017/09/01 by Krylov, Vasily
#FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.1709.00391

Abstract

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_λ₁,λ₂\colon B^G(λ₁) ⊗ B^G(λ₂) → B^G(λ₁+λ₂)∪\0\ 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\colonRep(G)\rightarrowRep(L) in terms of the hyperbolic localization functors for the generalized transversal slices.

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