Representations of Lie groups. Harmonic and complex analysis on symmetric and locally symmetric spaces

Research Project, 2019 – 2022

We will find branching rules of unitary representations of semisimple Lie groups and its applications to boundary value problems for conformally invariant differential operators. We study integral operators such as Knapp-Steinintertwining operators on differential forms and prove Hardy-Littlewood-Sobolev inequalities. We construct Shimura type invariant differential operators on general symmetric spaces and study the positivity of their spectral symbols. We introduce a new class of H-type fiberation over compact and non-compact symmetric spaces and study their sub-Riemannian geometry in connection with the realization complementary series of Lie groups. We study Dixmier trace of Toeplitz and Hankel operators. We classify isometric mappings between symmetric domains. We study Riemannian and Kaehler metrics on Hitchin components. We study variations of the analytic torsion, Selberg zeta functions on Riemann surfaces and related open question on Morse functions, convexity and extremal points. We plan to organize small workshops on representation theory, analysis on locally symmetric spaces and Hitchin components. We shall invite leading experts to Sweden to give lectures on new developments. We intend to work with young faculty members in Gothenburg to establish a theory for regularized determinants and Dixmier trace for the classical C alderon-Zygmund commutators, Hankel operators and also for pseudo-differential operators on locally symmetric spaces.