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

We study unitary representations of Lie groups and related harmonic and complex analysis on symmetric spaces, locally symmetric spaces and complex manifolds.

We study variational problems related to higher Teichmueller theory.We plan to find new unitary irreducible representations by studying Heisenberg parabolically induced representations. We study the branching of complementary series and their applications to fractional Laplacian operators.

We will prove the positivity of Euclidean Fourier transform of Harish-Chandra spherical functions.We will prove sharp integral estimates for matrix-coefficients for unitary highest weight representations. This can be viewed as a generalization of the Schur´s orthogonality relation and the Weyl dimension formula.

We study  tensor product decompositionsand Toeplitz  operators on Bergman spaces on compact and non-compact manifolds, and we shall find their applications to sharp estimates.We classify isometric holomorphic mappings between symmetric domains by using Bergman spaces of vector-valued holomorphic functions and Jordan triples.

We study Kähler metrics on Hitchin components.

We study variations of  energy functions on Teichmueller space and their convexity property.We plan to organize small workshops on representation theory, harmonic and complex analysis and Hitchin components. We shall invite leading experts to Sweden to give lectures on new developments.