Lorentzian and Octonionic Satake equivalence

Abstract

We establish a derived geometric Satake equivalence for the real group $G_{\mathbb{R}}=PSO(2n-1,1)$ (resp. $P{E}_6(F_4)$), to be called the Lorentzian Satake equivalence (resp. Octonionic Satake equivalence). By applying the real-symmetric correspondence for affine Grassmannians, we obtain a derived geometric Satake equivalence for the splitting rank symmetric variety $X=PS{O}_{2n}/S{O}_{2n-1}$ (resp. $P{E}_6/F_4$). As an application, we compute the stalks of the $\text{IC}$-complexes for spherical orbit closures in the real affine Grassmannian for $G_{\mathbb{R}}$ and the loop space of $X$. We show the stalks are given by the Kostka-Foulkes polynomials for $G{L}_2$ (resp. $G{L}_3$) but with $q$ replaced by $q^{n-1}$ (resp. $q^4$).