Analysis on semisimple Lie groups is a very vital branch of modern analysis. Its goal is to study unitary representations in Hilbert spaces and to relate differential and integral operators on homogeneous spaces. An abstract setting for studying representations in Banach spaces is via Banach group algebras on locally compact groups, known as abstract harmonic analysis. Analysis on Lie groups and abstract harmonic analysis are two closely related fields. One concerns more on the L2-theory, centering on the Plancherel formula, whereas the other concerns more on the functional analytic and algebraic structures of the Fourier algebras and some other related algebras. These two fields were studied together in the original works of Godement, Segal and Gelfand, and were diverted later on. However, recent developments indicated that it is important to combine the concrete Lie group analysis and the abstract Banach algebra techniques, in order to develop some systematic theory for representations of infinite dimensional Lie groups. In particular, the striking advancements for the branching problems made by, among many others, T. Kobayashi, B. Sun and C.-B. Zhu call for a new formulation in the setting of representations in Banach spaces.

We intend to bring some leading Chinese and overseas experts on these two subjects to Sanya, and to offer young Chinese mathematicans an opportunity to meet with the experts. We shall focus on the following topics and related problems.

(1) Dirac cohomology of representations of semisimple Lie groups and its application to branching rules, realization of quaternionic representations and their application to classical quaternionic analysis.

(2) Positive definite functions on Lie groups and their relation with probability theory and amenability problem in Banach algebras.

The workshop would provide a timely and stimulating venue for Chinese and foreign researchers to study the existing problems and to formulate new promising directions.

The field of representation theory of Lie groups is relatvely less well-developed in China. We shall thus invite more recently graduated Ph. Ds from mainland China. We believe this will foster more cooperation and exchange between the Chinese and overseas researchers in the field.