Ramification of Supercuspidal Parameters

Explore a comprehensive lecture on the ramification of supercuspidal parameters delivered by Michael Harris from Columbia University as part of the Theta Series: Representation Theory, Geometry, and Arithmetic conference. Delve into topics such as the local Langlands conjecture, Kaletha's parametrization, the Deligne-Kazhdan correspondence, and V. Lafforgue's global results. Examine concepts like incorrigible representations, globalization, wild ramification, and mixed supercuspidals. Gain insights into the application of purity, Poincaré series, and potential automorphy. This 58-minute talk provides an in-depth exploration of advanced mathematical concepts in representation theory and number theory.

Intro
Outline
No the series
What is the local Langlands conjecture?
First version of LLC
Automorphic conditions
Fargues-Scholze
Kaletha's parametrization
The Deligne-Kazhdan correspondence
An exercise
Review of V. Lafforgue's global results
Weights
What about supercuspidals?
Incorrigible representations
Globalization
Application of purity
Poincaré series
Wild ramification
Mixed supercuspidals
Assuming multiplicity one and stable basse change
An inductive proof
Application of potential automorphy