Geometry of nilpotent orbits and representation theory – NilpOrbRT

We combine different approaches: in Lie theory (structural properties of Lie algebras et their adjoint orbits), in algebraic geometry (properties of algebraic groups), and also in combinatorics. The combinatorial methods often turn out to be necessary for obtaining more precise results. However they require that one focuses on particular cases (at least on the classical cases).

A theorem has been obtained which establishes the existence of affine pavings in the classical cases for certain varieties which generalize the Springer fibers. This theorem generalizes a classical result due to De Concini, Lusztig, and Procesi.

It seems interesting to study whether the theorem mentioned in the previous section «Results« can be generalized in the setting of affine Springer fibers.
Other works in progress concern the study of Springer representations, in particular their induction in the setting of symmetric pairs.

A preprint entitled «Partial flag varieties and nilpotent elements« is available on the website arXiv (reference 1305.3355).

This proposal is concerned with a problem of fundamental mathematics. The topics of the proposal lie at the interface of the domains of representation theory, Lie theory, and complex algebraic geometry. More precisely the proposal focuses on interactions between these domains. The Lie algebra of a reductive algebraic group always contains a finite number of nilpotent orbits. For the last 35 years, the latter have turned out to play an important role in the representation theory of the group. The purpose of the project is to study certain geometrical aspects of the nilpotent orbits through their relation with representation theory. We will mainly consider two aspects: the study of the geometry of the orbital varieties and the study of the representations of the W-algebras. The orbital varieties are Lagrangian subvarieties of the nilpotent orbits, which can be interpreted as the geometrical analogues of the primitive ideals of the enveloping algebra. We will study different conjectures on the inclusion relations between orbital variety closures for the type A, in the light of recent results of Hinich and Joseph. A W-algebra is an associative algebra associated to the datum of the Lie algebra and any nilpotent element. Concerning this algebra, we will study the highest weight theory recently developed by Brundan, Goodwin, and Kleshchev. Finally, we will try to construct algebraic quantizations of W-algebras, which would be explicit and practical. To do this, we will be inspired by the construction of geometrical quantizations of W-algebras recently obtained by Losev.