Professor Gaillard will study the structure of Harish- Chandra modules for semisimple Lie groups. In particular he will study the relationship of the representation algebra and the extension algebra and try to show that they are dual formal algebras in the sense of Beilinson and Ginsburg, a conjecture he has verified already in special cases. This research involves the theory of group representations. Group theory is basically the study of symmetry. To take an example, when the system in question is invariant under a change in the position of the origin of space, the group of translations naturally arises. While groups are abstract objects, particular situations demand concrete realizations or "representations" of the symmetry group.
