Restriction Problems in Representation Theory

University Of Utah, Salt Lake City UT

Investigators

Abstract

A central theme in this project is Langlands program, which is a surprising mixture of analysis on groups and number theory. A basic example is the Fourier expansion of a function on the circle as a sum of trigonometric functions. The circle is a very symmetric object. We can rotate it for any angle without changing its shape. Rotations of a circle form what is called in mathematics a group of transformations, or simply a group. Every function on the circle has a Fourier expansion in which it is written as a sum of trigonometric functions. Trigonometric functions are the simplest functions on the circle, and so the analysis of functions on the circle, or periodic functions, is reduced to trigonometric functions. It has been known for over a century that Fourier expansions and their generalizations carry significant number theoretic information. Motivated by this example, over the past fifty years a large class of problems has been unified in a scheme known as the Langlands program. Roughly speaking, the goal is to decompose any function on a group as a sum of the simplest possible functions. This decomposition problem is the main theme of this project. This project will include training of graduate students. More specifically the main goal of this project is to complete the Langlands classification for the group G2. This group was discovered by Eugene Dickson and it has been found to play an important role in physics and string theory. The principal tool employed in this research is the theory of exceptional theta correspondences that was developed by the PI in previous work. Applications of the exceptional theta correspondence to dual pairs of real groups and to supercuspidal representations of p-adic groups will developed as well. This project will include training of graduate students. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.