Categories parametrized by schemes and representation theory in complex rank

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.