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

TL;DR

This paper explores how epsilon factors relate to the complex structure of the smooth dual of GL_n over a non-archimedean local field, revealing their factorization as invariant characters and providing explicit formulas in unramified cases.

Contribution

It demonstrates the interface between epsilon factors and the complex structure of the smooth dual, with explicit formulas for unramified cases, advancing understanding of local Langlands correspondence.

Findings

Epsilon factors factor as invariant characters through complex tori.

Explicit formulas for invariant characters in unramified cases.

Epsilon factors interface with the complex structure of the smooth dual.

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.