This mathematics research project by Wilfried Schmid consists of two loosely related parts. In collaboration with Kari Vilonen, Schmid will complete the proof of their recent conjecture about irreducible unitary representations of reductive Lie Groups. Classifying the irreducible unitary representations of such a group G is known to be equivalent to an algebraic problem: among the irreducible Harish-Chandra modules with an invariant, but possibly indefinite inner product, determine those for which the inner product is positive definite. Vogan and his collaborators on the AIM Atlas project have pointed out that this inner product is directly and explicitly related to a certain indefinite inner product, one that is infinitesimally invariant under a compact real form U of the complexification of G. According to the above-mentioned conjecture, the U-invariant inner product is computable in terms of Morihiko Saito's Hodge filtration on the Beilinson-Bernstein D-module realization of the Harish-Chandra module in question. The conjecture would not explicitly classify the irreducible unitary representations, but would bring the functorial apparatus of Hodge theory to bear on the unitarity problem. Schmid and Vilnoen will prove the conjecture, and also investigate its various implications. The other component of the project is joint with Steve Miller. Schmid and Miller and have developed a new method for proving the functional equations and holomorphy for Langlands L-functions. Compared to the existing methods, it has the advantage of making the Gamma factors directly computable, at least in all the cases we have examined so far. This enables them to exclude all unexpected poles of L-functions that the other methods cannot rule out. In principle, it should apply to all L-functions accessible by the method of integral representations. Miller and Schmid plan to refine their method and extend its range of applicability.
This mathematics research project by Wilfried Schmid is in the general area of representation theory, specifically on the representation of so-called non-compact groups of symmetries. Symmetry is a familiar phenomenon that occurs in everyday life. The concept of symmetry was formalized by 19th century mathematicians, who introduced the notion of "group of symmetries"; in this context the group of rotations of three dimensional space is a basic but important example because the laws of classical mechanics do not change under space rotations, and this fact helps to organize, simplify and thus better understand the solution of many practical problems in other disciplines such as relativity theory. While representations of compact groups have been well understood for three quarters of a century, the same is not true for the representations of non-compact groups, which are among the topics studied by this project. In addition to this work, Schmid is involved in several activities pertaining K-12 education, such as serving in advisory panels, and giving public lectures.
