[PDF] Chromatic aberrations of geometric Satake over the regular locus-论文阅读讨论-ReadPaper

摘要

Let $G$ be a connected and simply-connected semisimple group over$\mathbf{C}$, let $G_c$ be a maximal compact subgroup of $G(\mathbf{C})$, andlet $T$ be a maximal torus. The derived geometric Satake equivalence ofBezrukavnikov-Finkelberg localizes to an equivalence between a full subcategoryof $\mathrm{Loc}_{G_c}(\Omega G_c; \mathbf{C})$ and$\mathrm{QCoh}(\check{\mathfrak{g}}^{\mathrm{reg}}[2]/\check{G})$, which can bethought of as a version of the geometric Satake equivalence "over the regularlocus". In this article, we study the story when $\mathrm{Loc}_{T_c}(\OmegaG_c; \mathbf{C})$ is replaced by the $\infty$-category of $T$-equivariant localsystems of $A$-modules over $\mathrm{Gr}_G(\mathbf{C})$, where $A$ is acomplex-oriented even-periodic $\mathbf{E}_\infty$-ring equipped with anoriented group scheme $\mathbf{G}$. We show that upon rationalization,$\mathrm{Loc}_{T_c}(\Omega G_c; A)$, which was studied variously byArkhipov-Bezrukavnikov-Ginzburg and Yun-Zhu when $A = \mathbf{C}[\beta^{\pm1}]$, can be described in terms of the spectral geometry of variousLanglands-dual stacks associated to $A$ and $\mathbf{G}$. For example, thisimplies that if $A$ is an elliptic cohomology theory with elliptic curve $E$,then $\mathrm{Loc}_{T_c}(\Omega G_c; A) \otimes \mathbf{Q}$ can be describedvia the moduli stack of $\check{B}$-bundles of degree $0$ on $E^\vee$.

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