Branching for representations of semisimple Lie groups and automorphic forms

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This mathematics research project deals with the restriction of infinite dimensional unitary representations of a semisimple Lie group G to a semisimple subgroup H and applications to automorphic forms. The project also concerns the restriction of the underlying Harish Chandra-modules and their globalizations. The restrictions of unitary representations are not yet well understood and in particular understanding the restriction of the HarishChandra-modules is still in its infancy. So examples play a major role. Speh and her collaborators will also study the restriction of the HarishChandra-modules of the rank one orthogonal groups and to work out the restriction of complementary series representations, which are of particular analytic interest. These restriction problems have applications to automorphic forms and the cohomology of arithmetic groups, which Speh will also investigate; in particular she is interested in generalized modular symbols and period integrals defined by symmetric subgroups. This mathematics research project in the area of group representation theory deals in a fundamental way with the understanding of the symmetries of a space, such as the symmetries of an atom or the rotations of a sphere. Symmetries have important applications to several disciplines such as robotics and civil engineering. In robotics, the symmetries of the ``configuration space" are important and are limiting factors in the controllability of the arms of a robot. The stability of the impressive domes over some large buildings, such as sport stadiums, is due in part to the domes' symmetries. As part of this project, Speh will organize an interdisciplinary conference that will bring together senior and junior researchers from several disciplines.