Abstract Adams This project concerns representation theory of non-linear reductive groups, with an emphasis on number-theoretic applications. One focus will be on lifting of characters from linear to non-linear groups. This builds on work of Flicker, Kazhdan and others, and generalizes joint work with Jing-Song Huang on GL(n,R), and Adam's work on Mp(2n,R). Ultimately it may reduce the study of characters of non-linear groups to linear groups, which are more well understood, and thereby bring non-linear groups into the Langlands program. In joint work with Steve Rallis and Carey Rader, Adams will also look at a conjectural geometric matching of orbital integrals between SO(2n+1) and Mp(2n,F) over a p-adic or real field F. Finally Adams plan to continue his joint project with Dan Barbasch on the dual pair correspondence over R. Lie groups (named after the Norwegian mathematician Sophus Lie) are among the most ubiquitous objects in mathematics and science. Lie groups are the mathematical framework for the study of symmetry. They play an important role wherever symmetry is found, for example in the design of particle accelerators. Typically a Lie group is not seen directly, but appears via a "representation" of it, which gives rise to the study of representations of Lie groups, In recent years more exotic "non-linear" Lie groups have played an increasingly important role, although their representation theory is not as well understood as in the linear case. The main goal of this project is to bring the study of representations of non-linear Lie groups up to the level of linear Lie groups.
