Research

Research interest – Representation theory, algebraic groups, geometry of nilpotent orbits, Springer theory, combinatorics arising from representation theory.

The (generalised) Springer theory for reductive groups plays an important role in representation theory. It relates nilpotent orbits to irreducible representations of Weyl groups. In previous work [3,4,5,7,13] we describe the Springer correspondence for (classical) Lie algebras and their duals in bad characteristics. Many questions in algebraic groups have rather uniform answers over fields of large characteristic. When the characteristic of the base field is small, special interesting phenomena can occur. In [8,9,11] we study the nilpotent adjoint and coadjoint orbits in small characteristic and their properties. In particular we describe the nilpotent pieces (following Lusztig). These pieces have nice properties independent of the characteristic of the base field.

My current research focus is to establish a Springer theory in the setting of graded Lie algebras together with my collaborators. In [15,19] we develop such a theory for symmetric pairs and classify the character sheaves (Fourier transforms of simple equivariant perverse sheaves supported on the nilpotent cone) arising in this setting. This in turn relies on a key geometric construction [18] of nearby cycle sheaves. As one application, we obtain a strategy to compute cohomology of Hessenberg varieties. The strategy was carried out in [17] for a special class of Hessenberg varieties, certain moduli spaces of vector bundles with extra structure on a hyper-elliptic curve. In the context of graded Lie algebras, Hecke algebras at roots of unity associated to complex reflection groups enter the description of character sheaves. In [20] we describe (conjecturally all) full support character sheaves explicitly in the case of stably graded classical Lie algebras. We make use of the nearby cycle sheaf construction for stable polar representations established in [19].