Geometric Satake Equivalence: From Perverse Sheaves to Dual Group Representations — E8 Intelligence Research

FINDING: The geometric Satake equivalence establishes a tensor category isomorphism between perverse sheaves on the affine Grassmannian and representations of the Langlands dual group, with recent advances extending to derived categories, supergroups, and mixed characteristic. MATH: - Core objects: Affine Grassmannian \( Gr_G = G(C((t)))/G(C[[t]]) \) - Coweight lattice: \( X_*(T) = Hom(G_m, T) \) — the lattice of 1-parameter subgroups - Equivalence: \( PervG(O)(Gr_G) Rep({G}) \) (tensor categories) - Convolution structure: \( F_1 F_2 = p_!(q_1^*F_1 ⊗ q_2^*F_2) \) via the convolution diagram - Recent variants: derived categories \( D^b(Gr_G) \), supergroup analogues (Finkelberg), mixed characteristic (Scholze via v-stacks and diamonds) - Key constants: none explicitly, but the structure is governed by root system data Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com