Let $\mathcal Y$ be a derived algebraic stack satisfying some mild conditions. The purpose of this paper is three-fold. First, we introduce and study $\mathbb H(\mathcal Y)$, a monoidal DG category that might be regarded as a categorification of the ring of differential operators on $\mathcal Y$. When $\mathcal Y = \mathrm {LS}_G$ is the derived stack of $G$-local systems on a smooth projective curve, we expect $\mathbb H (\mathrm {LS}_g)$ to act on both sides of the geometric Langlands correspondence, compatibly with the conjectural Langlands functor. Second, we construct a novel theory of D-modules on derived algebraic stacks. In contrast to usual D-modules, this new theory, to be denoted by $\mathcal D^{\mathrm{der}}$, is sensitive to the derived structure. Third, we identify the Drinfeld center of $\mathbb H(\mathcal Y)$ with $\mathcal D^{\mathrm{der}}(L\mathcal Y)$, the DG category of $\mathcal D^{\mathrm{der}}$-modules on the loop stack $L\mathcal Y$: = $\mathcal Y \times_{\mathcal Y \times \mathcal Y}\mathcal Y$.

中文翻译：

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微分算子分类环的中心

令$ \ mathcal Y $是满足某些温和条件的派生代数堆栈。本文的目的是三方面的。首先，我们介绍并研究$ \ mathbb H（\ mathcal Y）$，这是一个单项DG类别，可以视为$ \ mathcal Y $上微分算子环的分类。当$ \ mathcal Y = \ mathrm {LS} _G $是平滑投影曲线上的$ G $-局部系统的派生堆栈时，我们期望$ \ mathbb H（\ mathrm {LS} _g）$都将作用于两边与几何Langlands函子兼容。其次，我们在导出的代数堆栈上构造了D-模块的新理论。与通常的D模块相反，用$ \ mathcal D ^ {\ mathrm {der}} $表示的这一新理论对导出的结构很敏感。第三，