Parabolic conjugation on nilpotent elements for classical Lie types

• Applicant: Dr. Magdalena Boos
• Subject Area: Mathematics
• Term: from 2018 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 409550143

Let G be a simple classical complex Lie group and let g be its Lie algebra. Let N be the nilpotent cone of nilpotent elements in g and denote by N(m) the subvariety of such nilpotent elements n of nilpotency degree m, that is, n^m=0. We consider the conjugation-action of an arbitrary standard parabolic subgroup of G on the variety N(m). Our main aim is to prove a criterion which lists all cases of parabolic subgroups and integers m for which the described action only has finitely many orbits. We want to understand the finite cases in detail, for example in terms of a parametrization of the orbits by combinatorial objects, by describing degenerations of orbits and by calculating singularities in orbit closures. In the infinite cases, we intend to describe infinite families of orbits and would like to define semi-invariants which generate the parabolic semi-invariant rings.By translating the setup to a representation-theoretic context in the language of finite-dimensional algebras by means of quivers with relations, many of the described questions have been answered in the case of G being the general linear group. A similar translation is possible for symplectic and orthogonal Lie types - here, symmetric representations of symmetric quivers with relations have to be considered, which made it possible to prove first results for the case that m=2. We aim to use this translation in order to prove our aspired results by expanding the so far known symmetric representation theory. The case where G is the general linear group gives many clues on how to proceed here.

• DFG Programme: Research Fellowships
• International Connection: Italy
• Host: Professor Dr. Giovanni Cerulli Irelli