Representation Theory in Complex Rank

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. The subject of representation theory in complex rank goes back to the papers [DM, De1]. Namely, these papers introduce Karoubian tensor categories Rep(GL t) ([DM, De1]), Rep(O t), Rep(Sp 2t), t ∈ C ([De1]), which are interpolations of the tensor categories of algebraic representations of classical complex algebraic groups GL n , O n , Sp 2n to non-integral rank 1. This means that when t = n is a nonnegative integer, these categories project onto the corresponding classical representation categories Rep(GL n), Rep(O n), Rep(Sp 2n), i.e., they have tensor ideals, the quotients by which are the classical representation categories. Later, in [De2], P. Deligne introduced Karoubian tensor categories Rep(S t), t ∈ C, which are similar interpolations for the representation category of the symmetric group Rep(S n) (and project onto it for t = n). In [Kn1, Kn2], F. Knop proposed a broad generalization of Deligne's construction. In particular, he generalized his construction for S n to the case of wreath product groups S n ⋉ Γ n , where Γ is any finite group, constructing Karoubian tensor categories Rep(S t ⋉ Γ t) for complex t, projecting for t = n onto Rep(S n ⋉ Γ n). Since these categories are semisimple for non-integer t, one may think of these results as " compact " representation theory in complex rank. The goal of this paper is to start developing the " noncompact " counterpart of this theory. Namely, in this paper we will introduce a method that allows one to define interpolations to complex rank of various categories of representations of classical type, in particular the following ones: 1) wreath products; 2) degenerate affine Hecke algebras; 3) rational and trigonometric Cherednik algebras, symplectic reflection algebras; 4) real groups (i.e., symmetric pairs); 1 In fact, Rep(O t) = Rep(Sp −t) with a modified symmetric structure. 5) Lie superalgebras; 6) affine Lie algebras; 7) Yangians; 8) (Parabolic) category O for reductive Lie algebras. Namely, we will define representations of a " noncompact algebra " of complex rank as representations of its " maximal compact subalgebra " (i.e. an (ind)-object of the corresponding tensor category) together with some additional structure (morphisms satisfying some relations). These morphisms and relations are obtained by writing down a " categorically friendly " definition of the …