A new way to construct \(1\)-singular Gelfand-Tsetlin modules

Abstract

We present a simplified way to construct the Gelfand-Tsetlin modules over
$\gl(n,\CC)$ related to a $1$-singular GT-tableau defined in
\cite{FGR-singular-gt}. We begin by reframing the classical construction of
generic Gelfand-Tsetlin modules found in~\cite{DFO-GT-modules}, showing
that they form a flat family over generic points of $\CC^{\binom{n}{2}}$. We
then show that this family can be extended to a flat family over a variety
including generic points and $1$-singular points for a fixed singular pair
of entries. The $1$-singular modules are precisely the fibers over these
points.