Abstract

In this two-part paper, we introduce a \(p\)-adic analytic analogue of Backelin and Kremnizer’s construction of the quantum flag variety of a semisimple algebraic group, when \(q\) is not a root of unity and \(\vert q-1\vert<1\). We then define a category of \(\lambda\)-twisted \(D\)-modules on this analytic quantum flag variety. We show that when \(\lambda\) is regular and dominant and when the characteristic of the residue field does not divide the order of the Weyl group, the global section functor gives an equivalence of categories between the coherent \(\lambda\)-twisted \(D\)-modules and the category of finitely generated modules over \(\widehat{U_q^\lambda}\), where the latter is a completion of the ad-finite part of the quantum group with central character corresponding to \(\lambda\). Along the way, we also show that Banach comodules over the Banach completion \( \widehat{ \mathcal{O}_q(B) } \) of the quantum coordinate algebra of the Borel can be naturally identified with certain topologically integrable modules.

中文翻译：

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解析量子组的Beilinson-Bernstein定理。一世

摘要

在这个由两部分组成的论文中，我们介绍了Backelin的\（p \） -adic解析类似物和Kremnizer构造的一个半简单代数群的量子标志变体，其中\（q \）不是统一性且\（ \ vert q-1 \ vert <1 \）。然后，在此解析量子标记变量上定义\（\ lambda \）-加捻的\（D \）-模块的类别。我们表明，当\（\ lambda \）是规则且占优势时，并且当残差字段的特征未划分Weyl基团的顺序时，全局截面函子给出了相干\（\ lambda \）之间类别的等价性缠绕\（D \）-modules和\（\ widehat {U_q ^ \ lambda} \）上有限生成的模块的类别，其中后者是量子群的ad-finite部分的完成，其中心字符对应于\（\ lambda \）。在此过程中，我们还证明了Borel量子坐标代数的Banach完备\（\ widehat {\ mathcal {O} _q（B）} \）上的Banach协模可以自然地由某些拓扑可积模块识别。