This proposal contains two separate strands, both connected to the representation theory of reductive Lie groups: the study of irreducible unitary representations of such groups, and functional equations of L-functions attached to automorphic representations. In collaboration with Kari Vilonen, Schmid shall apply M. Saito's theory of mixed Hodge modules to the study of irreducible unitary representations; the objective is to derive criteria for unitarity. Schmid will continue his collaboration with Steve Miller. They will prove the regularity of L-functions, and derive functional equations, from the point of view of automorphic distributions, in cases that are inaccessible by other methods. Classical Fourier analysis is an absolutely fundamental tool for the study of functions of one or several real variables. In the 20th century, Fourier analysis was extended to abelian groups, compact groups, and finally to non-compact, non-abelian groups. Irreducible unitary representations constitute the basic building blocks of Fourier analysis. Although Harish-Chandra constructed enough irreducible unitary representations carry out Fourier analysis on reductive Lie groups, all irreducible unitary representations are necessary for Fourier analysis on quotient spaces of reductive Lie groups. Many partial results exist, but they do not fit into a coherent, general picture. Schmid and Vilonen recently formulated a far-reaching conjecture on the unitarity problem. Their aim is to verify the conjecture and to explore its various implications. Riemann's zeta function encodes deep properties of prime numbers, and Dirichlet L-functions do the same for primes in abelian extensions of the rational numbers. These functions are regular, except for certain well understood poles, have Euler products, and satisfy functional equations. Conjecturally Langlands L-functions play an analogous role for general number fields. They are defined in terms of Euler products, but their analytic properties are not obvious from the de¯nition. Functional equations and regularity have been established in important special cases. Miller and Schmid expect their method to apply in several new cases, and also to simplify some existing arguments significantly.
Irreducible unitary representations of reductive groups play an important role in the Langlands program and other areas of number theory. The mathematical physics literature abounds with examples of irreducible unitary representation having various special properties. A more systematic approach to the classification of unitary representations would help to unify this aspect of mathematical physics. Vogan and his collaborators are working on a computer algorithm which would determine whether any specified representation is unitarizable. For some groups, for example E8, such an algorithm will severely strain existing computing facilities. The conjecture of Schmid and Vilonen has a direct bearing on this massive computational problem, by allowing to cut down signi¯cantly on the number of cases that need to be checked. L-functions occupy a central place in algebraic and analytic number theory, and are now of interest also to cryptographers. This makes significant new results about L-functions useful to a large group of mathematicians, well beyond the community of experts on L-functions.
In a broad sense, the project was devoted to the study of symmetry, which is a fundamental notion in mathematics. Symmetry is formalized by the notion of a group, in the case of this particular project, so-called non-compact, reductive Lie groups. The basic building blocks of symmetries encoded by such groups are their irreducible unitary representations. Although they have been studied intensively for more than sixty years, they are still not completely understood. One aspect of the project is a new approach towards understanding these representations, using tools from algebraic geometry. Together with my collaborator Kari Vilonen, I have formulated a conjecture which, when proved, will provide completely new insights into their structure. We have made considerable progress towards proving the conjecture, but more needs to be done. Algebraic number theory is one area of mathematics that uses irreducible unitary representations of reductive Lie groups as an important tool. The Riemann zeta function encapsulates deep properties of prime numbers. Analogously, Langlands L-functions govern the behavior of primes in number fields. These L-functions are attached to automorphic representations -- particular instances of irreducible unitary representations of reductive groups. My collaborator Stephen Miller and I have developed a new technique to study properties of L-functions. We have applied this technique to prove results that could not be obtained by existing techniques. In addition to my mathematical research, I have maintained an active interest in K-12 mathematics education. In particular, I have served on government advisory panels and have lectured about mathematics education to audiences of mathematicians, teachers, and mathematics educators.
