  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.
