[ Français ]

Categorification and geometric representation theory are two of the most active and exciting branches of modern representation theory and are gradually increasing their importance in the field. These approaches have led to the introduction of new and powerful tools and a deeper understanding of underlying structures. Conversely, representation theory has been used as an organizational and computational tool in geometry and category theory leading, for example, to powerful new invariants in geometry and topology. It has become increasingly clear that categorification is a broad mathematical phenomenon with wide-ranging applications. For example, understanding the categorical representation theory of affine Lie algebras led to a proof of Broué’s conjecture for symmetric groups, a purely representation theoretic statement. More generally, the categorification of such mathematical objects as quantum groups and Hecke algebras has given us a new understanding of the structure of these basic objects and their representation theory.

Many ideas in categorification are related to geometric methods in representation theory. In fact, geometrization (the geometric realization of some algebraic structure) is often a precursor to categorification. For example, constructions of natural bases (such as Lusztig’s canonical bases in quantum groups or the Kazhdan-Lusztig bases in Hecke algebras) with positivity and integrality properties are a central part of geometric representation theory. The categorifications that are suggested by such geometric constructions provide rich explanations for the existence of these bases. This field is moving forward rapidly and giving exciting results such as progress on special cases of Lusztig’s conjecture, new results on the representation theory of Cherednik algebras, the identification of Koszul gradings on Schur algebras, and progress on “odd categorification”.