Professor Speh proposes to work on several problems concerning the restriction of an irreducible representation of a reductive Lie group G to a symmetric subgroup H. She proposes to obtain a branching formula for the restriction of certain class of unitary representations to a symmetric subgroup. The restriction of complementary series representations of groups of real rank one to subgroups of the same type will also be considered. Both of these problems have applications to automorphic forms and the cohomology of arithmetic groups. She also proposes to continue to investigate generalized modular symbols and period integrals defined by symmetric subgroups. She proposes furthermore to investigate together with D. Barbasch the representations in the residual spectrum of type E.
Representations of reductive Lie groups are used to describe the symmetry of a physical system, a differential equation, a geometric object. or a problem in Number Theory. Professor Speh proposes to study the "breaking of the symmetry", i.e consider the problem for a different, usually smaller, symmetry group. Special cases of this problem have been consider in physics. The intellectual merit of this research is a better understanding of symmetries in nature and in mathematics.
