Representation theory in complex rank
Seminar Room 1, Newton Institute
The subject of representation theory in complex rank was started by P.Deligne, who defined complex rank analogs for the classical complex groups GL(N),O(N),and Sp(N), and (later) for the symnmetric group S_N. These are certain symmetric tensor categories of superexponential growth (i.e. non-Tannakian), in which the dimension N of the generating object is a generic complex number. Later F.Knop generalized the latter construction to a large class of finite groups (such as general linear groups over a finite field). For generic N, these categories are semisimple, so one may think of these results as "compact" representation theory in complex rank.
I will speak about "noncompact" representation theory in complex rank. The discussion will include classical real groups (i.e., classical symmetric pairs), degenerate affine Hecke algebras, rational Cherednik algebras, affine Lie algebras, Yangians, and so on. By definition, a representation of a "noncompact algebra" of complex rank is a representation of its "maximal compact subalgebra" (i.e. an (ind)-object of the corresponding tensor category) together with some additional structure (morphisms satisfying some relations). I will discuss the explicit form of such morphisms and relations in several special cases (such as rational Cherednik algebras).
This approach leads to a multitude of new interesting representation categories, which, in a sense, capture the phenomenon of "stabilization with respect to rank" in representation theory of classical groups and algebras.