Geometry and Representations of P-adic Groups

PI将从不同的角度研究由Bushnell-Kutzko发起的p-adic群的表示理论中的类型理论。首先，我们（与Julee Kim一起）想证明PI构造的对实际上是类型。 这是已知的，到目前为止，只有当预期的相应的惯性类组成的supercuspidalrepresentations。 其次，PI证明了这种构造中的紧开子群是自然光滑群方案的基础。 我们设想，应该有更多的代数几何结构对，可能是一个上同调实现的类型。 这将带来新的连接与其他部分的表示理论的几何/上同调方法使用。表示理论的p-adic集团是一部分的朗兰兹计划，哲学连接表示与数论和自守形式。 因此，它是后者的许多应用的基石。 类型理论是近10年来的新发展，它为这一问题的研究提供了新的思路。 但是这些类型是相当复杂的，只有在特殊情况下才能很好地理解。 这个项目将赋予更多的结构和意义的类型，使新的连接与数论及其应用。

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.