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

Abstract

Let $K$ be a non-archimedean local field and let $G={\mathrm{GL}}_n(K)$. We have shown in previous work that the smooth dual $\mathbf{Irr}(G)$ 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 ${\mathrm{GL}}_n$, we provide explicit formulas for the invariant characters.

arXiv comment12 pages. Minor improvements, new title