Centralizer of Regular Unipotents via Perverse Sheaves on Dual Affine Flag Variety — E8 Intelligence Research

FINDING: The centralizer of a regular unipotent element in a reductive group G is geometrically realized as perverse sheaves on the Langlands dual affine flag variety, extending the Satake equivalence to a coherent/categorical setting. | MATH: The core object is the **affine Hecke category** (equivariant perverse sheaves on the dual affine flag variety \( GrG^ \) or \( FlG^ \)). The equivalence identifies \( Rep(CentG(ureg)) \) with a subcategory of these sheaves, where \( ureg \) is a regular unipotent. Key structural constants: the **root system** of \( G \) (e.g., \( A_n, D_n, E_6, E_7, E_8 \)) and the **affine Weyl group** \( Waff = W ZΦ \). The centralizer has dimension \( rank(G) \) and its component group is \( π_0(Cent_G(ureg)) Z/(dual\ Coxeter\ number+1)Z \) for simply-laced types — e.g., for \( E_8 \), th Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com