The integral geometric Satake equivalence in mixed characteristic

Abstract:

Let $k$ be an algebraically closed field of characteristic $p$. Denote by $W(k)$ the ring of Witt vectors of $k$. Let $F$ denote a totally ramified finite extension of $W(k)[1/p]$ and $\mathcal {O}$ its ring of integers. For a connected reductive group scheme $G$ over $\mathcal {O}$, we study the category $\mathrm {P}_{L^+G}(Gr_G,\Lambda )$ of $L^+G$-equivariant perverse sheaves in $\Lambda$-coefficient on the Witt vector affine Grassmannian $Gr_G$ where $\Lambda =\mathbb {Z}_{\ell }$ and $\mathbb {F}_{\ell } \ (\ell \ne p)$, and prove that it is equivalent as a tensor category to the category of finitely generated $\Lambda$-representations of the Langlands dual group of $G$.