局部系统的形变与Eisenstein级数
Deformations of local systems and Eisenstein series
摘要：
设$X$是一条光滑完备曲线，$G$是一个约化群.在[BG]中，我们介绍了我们称之为“几何艾森斯坦级数”的对象。这是一个反常层$\bar{Eis}_E$（或者更确切地说是一个复合体），它位于$X$上的主$G$ -丛的模栈$Bun_G(X)$上，它附着在$X$上的局部系统$E$上，相对于环面$\check{T}$，朗兰兹对偶于嘉当子群$T\subset G$。In loc.前引书我们证明了在几何Langlands对应的意义下，$\bar{Eis}_E$对应于由$E$诱导的$\check{G}$ -局域系统。在本论文中，我们解决以下问题，建议由V. Drinfeld：什么是反常层$Bun_G(X)$，对应于普遍变形$E$作为一个本地系统的博雷尔子群$\check{B}\subset \check{G}$？我们证明，根据Drinfeld的猜想，所产生的反常层，如果经典，即，非紧化Eisenstein级数
英文摘要：
Let $X$ be a (smooth and complete) curve and $G$ a reductive group. In [BG] we introduced the object that we called "geometric Eisenstein series". This is a perverse sheaf $\bar{Eis}_E$ (or rather a complex of such) on the moduli stack $Bun_G(X)$ of principal $G$-bundles on $X$, which is attached to a local system $E$ on $X$ with respect to the torus $\check{T}$, Langlands dual to the Cartan subgroup $T\subset G$. In loc. cit. we showed that$\bar{Eis}_E$ corresponds to the $\check{G}$-local system induced from $E$, in the sense of the geometric Langlands correspondence. In the present paper we address the following question, suggested by V. Drinfeld: what is the perverse sheaf on $Bun_G(X)$ that corresponds to the universal deformation of $E$ as a local system with respect to the Borel subgroup $\check{B}\subset \check{G}$? We prove, following a conjecture of Drinfeld, that the resulting perverse sheaf if the classical, i.e., non-compactified Eisenstein series.
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