
SCIENTIFIC PROGRAMS AND ACTIVITIES 

May 29, 2024  
Workshop on the Representation Theory of padic Groups

Paul Mezo, Carleton University
The local Langlands' program
Local class field theory tells us that the Galois group of an abelian
padic 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 nonabelian 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 padic 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 13 of Springer's Corvallis article "Reductive Groups)
Basic theory of padic 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 padic 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.
BushnellKutzko'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):
Bernstein's "Le 'centre' de Bernstein
Bushnell and Kutzko's "Smooth representations of padic groups: structure theory via types"
Rumelhart's
notes based on the course by Bernstein
JiuKang Yu, Purdue University
BruhatTits theory and buildings
This minicourse is designed to guide the participants to study and
use BruhatTits 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 BruhatTits 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.