Geometric Methods in the Local Langlands Correspondance for p-adic Groups.

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This research program is part of the local Langlands program for p-adic groups. The ambitious long term aim of this research program is the development of a categorical local Langlands correspondence for p-adic groups. The Langlands program is one of the major themes of modern mathematics and consists of a series of conjectures spanning number theory, representation theory, and the theory of automorphic forms. The Langlands program conjectures a correspondence between automorphic representations and representations of Galois groups and includes studying both the existence and functorialities of this correspondence. The local Langlands correspondence for p-adic groups is the part of this program related to algebraic and Galois groups over local fields. Though this correspondence is known in many cases, a conjectural interpretation of the correspondence as a series of functors has not been made. Even a conjectural description of the correspondence in terms of functors would be significant as this would allow for more systematic treatments in those already established cases and allow for proofs which will translate to the remaining unresolved cases. Our approach builds off of ideas of David Vogan, who introduced into our context the equivariant derived category of sheaves with constructable cohomology on the moduli space of Langlands parameters.  This geometric category is used to form a bridge between modules for Hecke algebras (attached to automorphic representations) and Langlands parameters (which are in turn associated to Galois representations). The precise aim of our program is to work towards understanding this bridge as a series of functors. Though the ultimate objectives are ambitious they lead us naturally towards two major research directions. The first research direction concerns the development of an explicit and workable description of the aforementioned geometric category. Within this theme, we propose to develop effective computational techniques to work explicitly with these objects. A second research direction is to reformulate the many functorialities of the Langlands correspondence in terms of functors on these geometric categories. There are reasons to believe that in many cases what one finds in the geometric context is in fact easier to describe than our current descriptions of these functorialities. Aside from the contribution these two tasks would make to our theoretical understanding, this work has the added benefit of providing an alternative approach to performing actual computations in the Langlands program. The development of explicit tools for working with these conjectured functorialities then provides an important, and currently often lacking, ability to test, explore, refine and even correct our existing conjectures.