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.
The project is intended to investigate the interactions between representation theory and algebraic geometry, the former is a branch of mathematics that studies the symmetry and the latter a branch that studies geometry by algebraic equations. The first main outcome is a new interpretation of quantum groups, an object under intense investigation in representation theory, by sheaves on varieties attached to oriented graphs (quivers) in algebraic geometry. The second main outcome from this project is a unified scene of quantum groups with its two/multi-paramater and super cousins by showing that their associated module categories of some sort are equivalent, providing a solution to the longstanding problem of determining certain multiplicities of simple objects in a standard object in these two/multi-parameter and super quantum groups. Geometry implicitly plays as a guideline in obtaining this result as well, for example the second parameter in a two-parameter quantum group has an incarnation as the mixed structure in geometry. The third main outcome is a first construction of (modified) coideal subalgebras, a new type of quantum groups, via partial flags of classical type, which leads to the construction of canonical bases for these algebras. During the supported period, the PI worked with numerous graduate students from various universities, and mentored several undergraduate students including a summer internship with a minority. He supervised a postdoc, ran seminars regularly at his host institute and organized an AMS special session in his research area. He presented his supported research work at several occasions in conferences and workshops, disseminating the research results.
