Epsilon factors as algebraic characters on the smooth dual of $\mathrm {GL}_n$

Let $K$ be a non-archimedean local field and let $G = {\rm GL}_n(K)$. We have shown in previous work that the smooth dual ${\rm Irr}(G)$ admits a complex structure: in this article we show how the epsilon factors interface with this complex structure. The epsilon factors, up to a constant term, factor as invariant characters through the corresponding complex tori. For the arithmetically unramified smooth dual of ${\rm GL}_n$, we provide explicit formulas for the invariant characters.