Projective geometry, invariants and momentum

• Applicant: Dr. Valdemar Tsanov
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 282475916

The subject of study are geometric spaces with many symmetries. In particular, I study embeddings of such spaces and linear representations of the symmetry groups. It is well-known that the class of semisimple complex Lie groups is exactly the class of linear automorphism groups of homogeneous projective varieties. The latter are called flag varieties. The irreducible representations of a semisimple group on finite dimensional vector spaces are in one-to-one correspondence with the projective embeddings of its flag varieties. The group acts on the ring of polynomials on the vector space. The invariant polynomials form a subring - the fundamental object in invariant theory. The invariant ring is related to a momentum map, defined with respect to a maximal compact subgroup of the symmetry group. The interplay between representations, invariants, momentum and projective geometry of flag varieties defines the subject area of the first part of this project. In particular, there is a remarkable relation between linear spans in the representation space and convex hulls in the momentum image. This naturally leads to the notion of secant varieties to flag varieties. These are classical geometric objects, but have not been systematically studied in the context of momentum maps. I plan to pursue such a study. Another central topic in representation theory is the branching problem. Given an embedding of groups, one asks about decompositions of representations of the ambient group with respect to the subgroup. This can be interpreted in the context of Geometric Invariant Theory, generalizing the above classical setting. The branching problem has two aspects: local, where one studies saturation coefficients, and global, where one seeks a single object encoding the complete branching laws for a given embedding. Objects containing global asymptotic information on multiplicities have recently been introduced by Seppänen, in the form of Okounkov bodies for Hilbert quotients. Many of their properties are not known. An investigation on Hilbert quotients and the local-global interaction in branching laws is intended as an essential part of the project. The second part of the project is about real semisimple Lie groups. In this context momentum maps are replaced by gradient maps. I intend to study gradient maps on representation spaces, in relation to convexity and sphericity. Gradient maps also apply to the geometry of orbits of real forms in complex flag varieties. In particular, I focus on bisectors in Hermitian symmetric spaces. Those are related to discrete groups. I am interested in a particular type of hypergeometric groups constructed recently by Brav and Thomas.

• DFG Programme: Research Grants