SCIENTIFIC PROGRAMS AND ACTIVITIES
|June 26, 2019|
Workshop on the Representation Theory of p-adic Groups
Paul Mezo, Carleton University
The local Langlands' program
Local class field theory tells us that the Galois group of an abelian p-adic field extension is isomorphic to a quotient group of the underlying field. From this it follows that representations of the Galois group are related to representations of the underlying field. It is from this perspective that many refer to the local Langlands program as a sort of non-abelian class field theory. The goal of these lectures will be to describe the precise conjectured relationship between Galois groups on the one hand and representations of reductive p-adic groups on the other.
Recommended background preparation: Knowledge of (click links for example material)
Inverse limits (eg. Lang's Algebra)
Galois theory (eg: Rotman's Galois Theory)
Basic structure theory of reductive algebraic groups, including root systems (eg: Sections 1-3 of Springer's Corvallis article "Reductive Groups)
Basic theory of p-adic fields (eg: Ch II of Lang's Algebraic Number Theory)
Alan Roche, University of Oklahoma
The Bernstein center and types
The Bernstein decomposition describes the blocks of the category of smooth complex representations of a reductive p-adic group. In other words, it expresses this category as the direct product of certain indecomposable full subcategories, often called components. We will begin by studying this decomposition together with some related results which describe the components as module categories over appropriate algebras.
Bushnell-Kutzko's theory of types proposes that the components can be analysed via certain special smooth irreducible representations of certain compact open subgroups. We will examine the basic definitions and constructions of this theory, and study how it relates to parabolic induction and Plancherel measure. Finally, we will look at some cases in which the objects posited by the theory have been constructed.
Recommended references (copies will be available at the workshop):
Jiu-Kang Yu, Purdue University
Bruhat-Tits theory and buildings
This mini-course is designed to guide the participants to study and use Bruhat-Tits theory in representation theory. The basic reference for this theory is Tits' article in the Corvallis proceedings. However, it was dated before 4 of the series of 5 papers (total of 550 pages) of Bruhat-Tits were written, and more recent applications in representation theory require some results in those technical papers that were not emphasized in Tits' article. We hope to provide a complement to Tits' article, and to include more recent developments.
Recommended background preparation:
Some familiarity with the contents of Tits' Corvallis article "Reductive groups over local fields" is assumed.