On the geometric Satake equivalence for Kac-Moody groups

TL;DR

This work extends geometric Satake to affine Kac-Moody groups by treating the double affine Grassmannian $\mathrm{Gr}_{G_{\mathrm{aff}}}$ as an $\infty$-stack and constructing a $t$-structure on $G_{\mathrm{aff}}[s]$-equivariant sheaves. The authors establish an abelian semisimple category equivalence with $\mathrm{Ind}(\mathrm{Rep}(G^{\vee}))$, and show that IC complexes correspond to irreducible highest-weight modules of the Langlands dual group $G^{\vee}$. A key methodological pillar is Braden-type hyperbolic localization, adapted to the KM setting, together with dimension estimates for affine MV-cycles to prove $t$-exactness of the constant term functor. The resulting KM geometric Satake is a significant step toward a KM Langlands program, providing a robust framework for relating equivariant perverse sheaves on KM affine Grassmannians to representation theory of the Langlands dual KM group. The work also outlines the potential extension to broader KM types and characteristic settings, and it sets the stage for future monoidal enhancements of the equivalence.

Abstract

This article establishes a geometric Satake equivalence for affine Kac-Moody groups as an equivalence of abelian semisimple categories over algebraically closed fields. We define a well-behaved category of equivariant sheaves on the double affine grassmannian \Gr_{G}, seen as a infty-stack, that we equip with a t-structure. We obtain an Braden's hyperbolic localization theorem for such a stack and prove that the constant term functor is t-exact using dimension estimates for affine MV-cycles. We then deduce the sought-for equivalence and prove that the IC-complexes match with the irreducible highest weight representations of the Langlands dual group G^{\vee}.

Theorems & Definitions (292)

• Theorem 1.2.1
• Theorem 1.4.1
• Theorem 1.6.1
• Theorem 1.6.2
• Definition 2.1.1
• Remark 2.1.2
• Lemma 2.1.3
• proof
• Lemma 2.1.4
• proof