Representations of Lie Groups and Supergroups

Abstract

The workshop focussed on recent developments in the representation theory of group objects in several categories, mostly finite and infinite dimensional smooth manifolds and supermanifolds. The talks covered a broad range of topics, with a certain emphasis on benchmark problems and examples such as branching, limit behavior, and dual pairs. In many talks the relation to physics played an important role. Mathematics Subject Classification (2000): 22E, 32M, 46G20 53D, 58B, 58C. Introduction by the Organisers The workshop Representations of Lie groups and supergroups was organized by Joachim Hilgert (Paderborn), Toshiyuki Kobayashi (Tokyo), Karl-Hermann Neeb (Erlangen), and Tudor Ratiu (Lausanne). From the very beginning applications in physics were a major motivation for the study of representations of Lie groups. Later also number theory, specifically the Langlands program, became a driving force. A lot of effort has been invested in the classification of unitary representations during the last two decades of the 20th century. There is a huge body of information available, but the central classification problems are still not completely solved. At the moment research in that direction is concentrated with some American research teams. The majority of research nowadays is focused on benchmark problems and examples, such as branching, limit behavior, and dual pairs. Moreover, the extension of the scope of representation theory to infinite dimensional groups on the one hand, and supergroups on the other, plays an important role. A common feature 736 Oberwolfach Report 13/2013 of these efforts is the identification of relevant classes of examples which are general enough to be interesting, but at the same time have enough restrictions to allow a general theory. In many cases the choice of these benchmark examples is guided by problems from theoretical physics. The focus of this workshop was on these recent developments. The meeting was attended by 51 participants from many European countries, Canada, the USA, and Japan. The meeting was organized around a series of 23 lectures each of 50 minutes duration. The set of speakers chosen was a mix of researchers in all stages of their careers, from very promising young post-docs to senior scientists who have been contributing key results to the field over the last 45 years. We feel that the meeting was exciting and highly successful. The quality of the lectures was outstanding and the intensity of discussions was exceptional even for Oberwolfach standards. A good indicator for this observation is the fact that all the available blackboards were occupied by discussion groups until late every evening of the meeting. What is even more remarkable is that the composition of these discussion groups changed every day. In particular, the researchers who have been focussing on either finite dimensional, infinite dimensional, or super contexts in their recent research did not stay among themselves. New collaborations have been started, and established research partners from different continents had the opportunity to discuss further projects in person. Without going too much into detail, let us mention some important new developments. In the area of infinite-dimensional Lie groups things are moving on two mutually interacting levels, one is the analytic theory of unitary representations and the other deals with geometric structures (symplectic, Poisson etc.) on manifolds with group actions. Based on new systematic approaches to specific classes of representations, we have seen precise classification results for various classes of groups such as oscillator groups, gauge groups and diffeomorphism groups (Janssens, Goldin, Zellner). Particulary interesting new directions are concerned with the combination of methods from stochastic analysis and quantum field theory with Lie theory (Gordina, Jorgensen, Vershik) and it also appears that, for certain classes of infinite-dimensional Lie algebras the global categorical perspective can provide deep new insights (Penkov). On the geometric side the powerful method of dual pairs is now emerging for important classes of infinite dimensional Hamiltonian systems (Gay-Balmaz), invariant theory for gauge groups is connected to singularity theory (Iohara) and new regularity results for differential equations on infinite dimensional groups have been obtained (Glockner). The analytic representation theory of Lie supergroups (as opposed to the algebraic representation theory of Lie superalgebras, which is also a thriving field but was not within the scope of this workshop) has made substantial progress in recent years, fueled in particular by questions of harmonic analysis originating from mesoscopic physics. Through this development a rapprochement of the Representations of Lie Groups and Supergroups 737 representation theory of supergroups and traditional representation theory of Lie groups can be observed. A similar effect can be observed for the interplay between representations of supergroups and Clifford analysis (Alldridge, de Bie, Przebinda, Wurzbacher). The main focus of the representation theory of finite dimensional Lie groups has shifted from the classification problem of the unitary dual of reductive groups (which is still unsolved) to structural results of representations such as branching problems to subgroups, and analysis on minimal representations of reductive groups. The interactions of infinite dimensional representations with global analysis on non-compact manifolds have been also actively studied, which often bring us new geometric insights. Interesting progress includes generalized Cartan decompositions for visible actions on complex manifolds (A. Sasaki) and for real spherical varieties (B. Krotz), analysis on branching laws to non-compact subgroups, and conformally equivariant differential systems (T.Kubo). More specific information is contained in the abstracts which follow in this volume. Representations of Lie Groups and Supergroups 739 Workshop: Representations of Lie Groups and Supergroups