Lorentzian and Octonionic Satake equivalence

TL;DR

This paper establishes new derived geometric Satake equivalences for specific real groups and symmetric varieties, connecting affine Grassmannian geometry with representation theory and computing stalks of IC complexes using Kostka-Foulkes polynomials.

Contribution

It introduces Lorentzian and Octonionic Satake equivalences for real groups and symmetric varieties, expanding the scope of geometric Satake theory.

Findings

Derived equivalences for $PSO(2n-1,1)$ and $PE_6(F_4)$.

Computed stalks of IC complexes using Kostka-Foulkes polynomials.

Connected affine Grassmannian geometry with representation theory.

Abstract

We establish a derived geometric Satake equivalence for the real group $G_{\mathbb R}=PSO(2n-1,1)$ (resp. $PE_6(F_4)$), 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 $X=PSO_{2n}/SO_{2n-1}$ (resp. $PE_6/F_4$). 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 $GL_2$ (resp. $GL_3$) but with $q$ replaced by $q^{n-1}$ (resp. $q^4$).