My reserach interests are in Lie theory, representation theory and invariant theory of linear algebraic groups.
In 2007 Helmut Strade and I have successfully completed our long-term project aimed at classifying all finite dimensional simple Lie algebras over algebraically closed fields of characteristic p>3 (for p>7 such a classification was obtained earlier in a series of papers by Block-Wilson and Strade). Our main Classification Theorem reads:
any finite dimensional simple Lie algebra over an algebraically closed field of characteristic p>3 is up to isomorphism either classical or a filtered Lie algebra of Cartan type or a Melikian algebra of characteristic 5.
There are still many interesting open problems in the classification theory of modular Lie algebras. As an example, J.-M. Bois used our Classification Theorem to prove that every finite dimensional simple Lie algebra over an algebraically closed field of characteristic p>3 is generated by two elements. This result is analogous to one of the well-known theorems on finite simple groups.
In 1995 I proved the Kac-Weisfeiler conjecture on p-divisibility of dimensions of simple modules over reductive Lie algebras of good characteristic. Modular representation theory of reductive Lie algebras is now a very active area of research. It is related with the geometry of Springer fibres via the derived version of the Beilinson-Bernstein equivalence discovered by Bezrukavnikov-Mirkovic-Rumynin. There are many open problems here and the interplay between representation theory, geometry of nilpotent orbits and Noetherian ring theory makes this area very attractive for young reserachers.
Around 2004 I started exploring a link between modular representation theory of reductive Lie algebras and the theory of primitive ideals of universal enveloping algebras over complex numbers. I came across this link when studying certain noncommutative deformations of Slodowy slices. These deformations arise as the endomorphism algebras of generalised Gelfand-Graev modules, but it turned out later that they are also isomorphic to finite W-algebras of mathematical physics (this was first rigorously proved in 2007 by D'Andrea-De Concini-De Sole-Heluani-Kac).
Representation theory of finite W-algebras is extremely rich and interesting. For example, it would be important for the theory of primitive ideals to prove that every finite W-algebra admits one-dimensional representations. As a partial confirmation of this, I proved in 2007 that every finite W-algebra has finite-dimensional representations (later this was reproved by Losev, who used a completely different method involving Fedosov quantisation).
One can attach a finite W-algebra to any nilpotent orbit in a complex semisimple Lie algebra, and all finite W-algebras have nice filtrations and PBW bases. However, defining relations of finite W-algebras are difficult to determine. For general linear Lie algebras this was done by Brundan-Kleshchev, who identified the finite W-algebras of type A with truncated shifted Yangians. Outside type A the presentation problem for finite W-algebras remains wide open.
More recently I got interested in the problem of classifying the maximal subalgebras of Lie algebras of simple algebraic groups defined over fields of positive characteristics. In the case of good positive characteriscs this problem has now been completely solved for exceptional groups in my joint paper with David Stewart which was published in Journal of The Americal Mathematical Society.
Apart from the above-mentioned topics I am also interested in various inter-related problems of Lie Theory involving invariant theory of algebraic groups. The keywords here are nilpotent orbits, commuting varieties, symmetric invariants of centralisers in reductive Lie algebras, and quantisations of Mishchenko-Fomenko subalgebras.