My research area is called Geometric Representation Theory. I enjoy thinking primarily about aspects of representation theory that can be "seen" geometrically.
I am particularly interested in the following topic:
Hecke algebras and their categorification via Soergel bimodules
Flag manifolds and the geometry of Schubert varieties
Affine Grassmannians and Geometric Satake Correspondence
Quantum groups and KLR algebras
Nilpotent Orbits and Springer Correspondence
Representations of Algebraic Groups
The Néron-Severi Lie Algebra of a Soergel module To appear in Transformation Groups
Following Looijenga and Lunts, we introduce for any Soergel module a semisimple Lie algebra generated by all the Lefschetz operators and their adjoints.
This algebra, called the Néron-Severi Lie algebra, can be used to provide a simple proof of the well-known fact that a Schubert variety is
rationally smooth if and only if its Betti numbers satisfy Poincaré duality.
We also determine a large set of elements, for finite Coxeter group, where the Neron-Severi Lie algebra coincides with the full Lie algebra
of endomorphism preserving the intersection form.
The Hard Lefschetz Theorem in positive characteristic for the Flag Varieties To appear in International Mathematical Research Notices We show that the Hard Lefschetz theorem holds for the flag variety of some reductive group in characteristic p, if p is larger than the number of positive roots.
The converse also holds, a part from 3 exceptional cases. One can also deduce an algebraic proof of the Hard Lefschetz theorem in the already known
characteristic 0 case.