Branching for representations of semisimple Lie groups and automorphic forms

本数学研究计画探讨半单李群G的无穷维酉表示对半单群H的限制及其在自守形式上的应用。该项目还涉及基本Harish Chandra模块及其全球化的限制。酉表示的限制还没有被很好地理解，特别是理解HarishChandra模的限制仍然处于起步阶段。因此，例子起着重要作用。斯佩和她的合作者还将研究限制的HarishChandra模块的秩一正交群，并制定出限制的互补系列表示，这是特别的分析兴趣。这些限制问题的应用自守形式和上同调的算术群，斯佩也将调查，特别是她感兴趣的是广义模块化符号和周期积分定义的对称子群。这个在群表示理论领域的数学研究项目涉及对空间对称性的基本理解，例如原子的对称性或球体的旋转。对称性在机器人和土木工程等多个学科中有重要的应用。 在机器人技术中，“位形空间”的对称性是重要的，并且是机器人手臂可控性的限制因素。在一些大型建筑物上，如体育场馆，令人印象深刻的圆顶的稳定性部分是由于圆顶的对称性。作为该项目的一部分，Speh将组织一次跨学科会议，汇集来自多个学科的高级和初级研究人员。

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.