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 \( \mathrm{Gr}_G = G(\mathbb{C}((t)))/G(\mathbb{C}[[t]]) \) - Coweight lattice: \( X_*(T) = \mathrm{Hom}(\mathbb{G}_m, T) \) — the lattice of 1-parameter subgroups - Equivalence: \( \mathrm{Perv}_{G(\mathcal{O})}(\mathrm{Gr}_G) \simeq \mathrm{Rep}(\check{G}) \) (tensor categories) - Convolution structure: \( \mathcal{F}_1 \star \mathcal{F}_2 = p_!(q_1^*\mathcal{F}_1 \otimes q_2^*\mathcal{F}_2) \) via the convolution diagram - Recent variants: derived categories \( D^b(\mathrm{Gr}_G) \), 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