REPRESENTATION THEORY IN COMPLEX RANK, I

Abstract

P. Deligne defined interpolations of the tensor category of representations of the symmetric group S n to complex values of n. Namely, he defined tensor categories Rep(S t ) for any complex t. This construction was generalized by F. Knop to the case of wreath products of S n with a finite group. Generalizing these results, we propose a method of interpolating representation categories of various algebras containing S n (such as degenerate affine Hecke algebras, symplectic reflection algebras, rational Cherednik algebras, etc.) to complex values of n. We also define the group algebra of S n for complex n, study its properties, and propose a Schur-Weyl duality for Rep(S t ).