P-adic Representation Theory and Geometry of the Lubin-Tate Tower

Regents Of The University Of Michigan - Ann Arbor, Ann Arbor MI

Investigators

Abstract

The principal investigator studies two outstanding problems in p-adic representation theory. The first problem is analyzing a conjectural geometric construction of supercuspidal representations of p-adic groups proposed by Lusztig in 1979. Lusztig's construction can be viewed as a special case of automorphic induction. While automorphic induction has been constructed in many special cases, the existing approaches are quite complicated and often rely on global methods. Lusztig's construction is much more elegant, but the questions of formalizing it and comparing it to the more classical constructions of p-adic representation theory remain completely open. It is expected that the results obtained by the principal investigator will shed light on the geometry that underlies the known cases of automorphic induction. The second problem was formulated by M. Harris in 2002. It asks for a construction of Bushnell-Kutzko types for the general linear group of a local field K in the cohomology of suitable analytic subspaces of the Lubin-Tate tower of K. The principal investigator studies a family of open affinoids in the Lubin-Tate tower whose cohomology is expected to realize various special cases of the local Langlands and Jacquet-Langlands correspondences (which will partially answer Harris's question). The relevant cohomology computations have much in common with the examples of p-adic Lusztig induction that have so far been understood. The Langlands Program has dominated much of research in algebra during the last 40 years. It has connections to some of the most prominent results in number theory and other areas of mathematics, such as Fermat's Last Theorem. The principal investigator works in a branch of this field known as the local Langlands program. It is concerned with the representation theory of the so-called p-adic groups, and the main driving force is the search for a general proof of the local Langlands correspondence. Various special cases of this correspondence have been obtained by Henniart, Harris, Taylor and many other mathematicians. However, most of the existing proofs are not explicit and do not provide sufficient information for the desirable applications of the local Langlands correspondence. The principal investigator uses methods of geometric representation theory for unipotent groups, developed in his previous works, to give new explicit constructions of the local Langlands correspondence, and to simplify and clarify the existing ones. One of the main tools is a conjectural geometric construction of representations of p-adic groups formulated by Lusztig in 1979. Until recently this construction has remained relatively unknown because it was not clear how to compare it to the more classical constructions. The principal investigator developed general techniques for analyzing this construction, and is currently using it to shed light on the geometry behind the local Langlands correspondence.