Representation theory and geometry of varieties associated to quivers

本文主要研究表征理论与颤振变体之间的相互作用。主要致力于在量子群（或量子化包络代数）中建立与仿射Hecke代数中相似互易的几何朗兰兹互易。为了实现这一目标，首席研究员建议通过某些颤振变体的几何来研究量子化包络代数表示理论中几个密切相关的基本问题，包括量子修饰代数及其正则基的几何实现，D型和仿射A型量子修饰代数的各种几何实现的比较，以及新一类颤振变体的发展。他还将研究量子修饰代数的代数范畴与几何范畴之间的关系，Schur代数的正则基与半正则基之间的关系，以及双参数量子群的几何构造。提出的活动将加强李论，代数几何和有限维代数理论之间的深层联系。这些问题的解决将最终导致量子群中几何朗兰兹互易的解决，并在此过程中证明量子修饰代数正则基上的正猜想。这个猜想二十多年来一直没有定论。建议的活动可能会在几何表示理论和代数几何方面取得进展，并将有助于进一步了解组合，图形和分类表示理论和其他数学领域。在提案期间，PI打算在其主办机构指导不同层次的本科生和研究生。他计划在他的主办机构组织研讨会和会议。该提案的研究成果将在PI计划访问的各个大学和PI将参加的会议上进行展示。合作，发表论文，有时新想法的灵感将在提案期间完成。

This proposal is concerned with the interactions between representation theory and quiver varieties. It is mainly dedicated to the establishment of geometric Langlands reciprocity in quantum groups (or quantized enveloping algebras) in analogy with similar reciprocity in affine Hecke algebras. To accomplish this goal, the principal investigator proposes to study several, closely related, fundamental problems in the representation theory of quantized enveloping algebras via the geometry of certain quiver varieties, including geometric realizations of quantum modified algebras and their canonical bases, the comparisons of various geometric realizations of quantum modified algebras of type D and of affine type A, and the development of new classes of quiver varieties. He will also investigate the relationship between algebraic and geometric categorifications of quantum modified algebras, the relationship between canonical bases and semicanonical bases of Schur algebras and the geometric construction of two-parameter quantum groups.The proposed activities will strengthen the deep connections between Lie theory, algebraic geometry and the theory of finite dimensional algebras. The solutions to the proposed problems will eventually lead to the settlement of geometric Langlands reciprocity in quantum groups and, along the way, the proof of the positivity conjecture on canonical bases of quantum modified algebras. This conjecture has been remained open for over twenty years. The proposed activities are likely to make progress in geometric representation theory and algebraic geometry and will help further knowledge in combinatorial, graphical and categorical representation theories and other fields of mathematics. During the time of the proposal, the PI intends to mentor undergraduate and graduate students of various levels at his host institute. He plans to organize seminars and conferences in his host institute. The research results in this proposal will be presented in various universities the PI plans to visit and conferences the PI will attend. Collaborations, published papers, and sometimes inspirations of new ideas will be accomplished during the time of the proposal.