Representation theory and geometry of varieties associated to quivers

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.