The proposal aims to apply methods from algebraic geometry to study representations of algebraic groups. The PI and his collaborators will further develop his theory of global Springer representations, which involves the geometry of Hitchin integrable systems, and apply it to representations of p-adic groups. The proposal also proposes to use geometric methods of Laumon, Ngo, etc. to prove orbital integral identities (Fundamental Lemmas) in relative trace formulae and to construct explicit automorphic forms (sheaves) for function fields and their Hecke eigenvalues (local systems), especially those with wild ramifications. e.g., Kloosterman sheaves for general reductive groups. Finally the proposed project also plans to study Koszul duality patterns for sheaves on generalized flag varieties.
This project naturally sits at the intersection of algebraic geometry, representation theory, number theory, and mathematical physics. In these subjects as well as in understanding our physical world, symmetry is a central theme. Group theory is a uniform way to study such symmetries, and representation theory is trying to classify the actions of groups on vector spaces. As often happens, the most interesting representations come from geometric objects with symmetries, which in turn appear in physics. This is why geometric methods are so powerful in solving representation-theoretic problems. Through this project, the PI hopes to shed light on hard problems in representation theory and number theory, and to discover more symmetries that appear in geometry.