Salvato in:
| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2409.03969 |
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Sommario:
- 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. $q^4$).