Categories parametrized by schemes and representation theory in complex rank

TL;DR

This paper develops an algebro-geometric framework to analyze categories parametrized by complex numbers, proving properties like semisimplicity and classifying simple objects for generic complex parameters, extending classical representation theory results.

Contribution

It introduces a new geometric method to study parameterized categories, enabling proofs of properties for generic complex parameters based on integer cases.

Findings

Deligne's categories are generically semisimple.

Finite-dimensional simple objects at transcendental parameters are quotients of standard objects.

The method applies to extrapolations of categories related to wreath products and affine Hecke algebras.

Abstract

Many key invariants in the representation theory of classical groups (symmetric groups $S_n$, matrix groups $GL_n$, $O_n$, $Sp_{2n}$) are polynomials in $n$ (e.g., dimensions of irreducible representations). This allowed Deligne to extend the representation theory of these groups to complex values of the rank $n$. Namely, Deligne defined generically semisimple families of tensor categories parametrized by $n\in \mathbb{C}$, which at positive integer $n$ specialize to the classical representation categories. Using Deligne's work, Etingof proposed a similar extrapolation for many non-semisimple representation categories built on representation categories of classical groups, e.g., degenerate affine Hecke algebras (dAHA). It is expected that for generic $n\in \mathbb{C}$ such extrapolations behave as they do for large integer $n$ ("stabilization"). The goal of our work is to provide a technique to prove such statements. Namely, we develop an algebro-geometric framework to study categories indexed by a parameter $n$, in which the set of values of $n$ for which the category has a given property is constructible. This implies that if a property holds for integer $n$, it then holds for generic complex $n$. We use this to give a new proof that Deligne's categories are generically semisimple. We also apply this method to Etingof's extrapolations of dAHA, and prove that when $n$ is transcendental, "finite-dimensional" simple objects are quotients of certain standard induced objects, extrapolating Zelevinsky's classification of simple dAHA-modules for $n\in \mathbb{N}$. Finally, we obtain similar results for the extrapolations of categories associated to wreath products of the symmetric group with associative algebras.