Representation theory and geometry of varieties associated to quivers

这一建议涉及表征理论和箭形变种之间的相互作用。它主要致力于建立量子群(或量子化包络代数)中的几何朗兰兹互易，类似于仿射Hecke代数中的类似互易。为了实现这一目标，主要研究者建议通过某些箭图簇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，包括量子修正代数及其标准基的几何实现，D型量子修正代数和A型仿射修正代数的各种几何实现的比较，以及新类箭图簇的发展。他还将研究量子修正代数的代数范畴和几何范畴之间的关系，Schur代数的标准基和半正则基之间的关系，以及双参数量子群的几何构造。所提出问题的解决将最终导致解决量子群中的几何朗兰兹互易，并在此过程中证明量子修正代数的正则基上的正性猜想。20多年来，这一猜测一直悬而未决。拟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、图形和范畴表示理论以及其他数学领域。在提议期间，PI打算在他所在的学院指导不同级别的本科生和研究生。他计划在他所在的学院组织研讨会和会议。这项提案中的研究成果将在该协会计划访问的各个大学和该协会将参加的会议上展示。合作、发表的论文，有时还有新想法的灵感，都将在提案期间完成。

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.