Globalization of supercuspidal representations over function fields and applications

Lecture 1 Wee Teck GAN

Course Description
I will describe how one can use Poincare series to construct cuspidal automorphic representations of connected reductive group over a global function field with a given supercuspidal local component and prescribed behavior (such as ramification) at all other places. This globalization result can be used to complete the Langlands-Shahidi theory over function fields (a recent result of Luis Lomeli). Combining this with the recent construction by Vincent Lafforgue of the global Langlands correspondence over function fields, one can prove stability of arbitrary Langland-Shahidi gamma factors and obtain the local Langlands correspondence for classical groups. I will discuss some of these applications and if time permits (for me to understand it), I will give a sketch of the Lafforgue’s construction.

Principal topics: Stokes equations; reversibility and the “scallop theorem”; fundamental singularities of Stokes flows (stokeslets, stresslets, rotlets); the Lorentz reciprocal theorem; Goursat representation of biharmonic fields in terms of analytic functions; conformal mapping; free boundary problems; mixed boundary value problems; transform techniques; boundary integral methods.