Geometry and Representations of P-adic Groups

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The PI will study the theory of types in representation theory ofp-adic groups, intiated by Bushnell-Kutzko, from different angles.Firstly, we (jointly with Julee Kim) would like to show that the pairsconstructed by the PI are actually types. This is known so far only whenthe expected corresponding inertia classes consist of supercuspidalrepresentations. Secondly, the PI has shown that the compact opensubgroup in this construction underlies a natural smooth group scheme. Weenvisage that there should be more algebrao-geometric structure on thepair, possibly a cohomological realization of types. This will bringnew connections with other parts of representation theory wheregeometric/cohomological method is used.The representation theory of p-adic groups is part of the Langlandsprogram, a philosophy linking representations with number theory andautomorphic forms. Hence it is the corner stone of many applications ofthe latter. The theory of types is a new development in the past 10 yearsand sheds new lights on the subject. But the types are rather complicatedobjects and are well-understood only in special cases. This project willendow more structures and meanings to the types to enable new connectionswith number theory and its applications.