The PI will study the theory of types in representation theory of p-adic groups, intiated by Bushnell-Kutzko, from different angles. Firstly, we (jointly with Julee Kim) would like to show that the pairs constructed by the PI are actually types. This is known so far only when the expected corresponding inertia classes consist of supercuspidal representations. Secondly, the PI has shown that the compact open subgroup in this construction underlies a natural smooth group scheme. We envisage that there should be more algebrao-geometric structure on the pair, possibly a cohomological realization of types. This will bring new connections with other parts of representation theory where geometric/cohomological method is used.
The representation theory of p-adic groups is part of the Langlands program, a philosophy linking representations with number theory and automorphic forms. Hence it is the corner stone of many applications of the latter. The theory of types is a new development in the past 10 years and sheds new lights on the subject. But the types are rather complicated objects and are well-understood only in special cases. This project will endow more structures and meanings to the types to enable new connections with number theory and its applications.
