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

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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.

Bibliographical metadata

Original languageEnglish
Title of host publicationProceedings of conference on Quantum Dynamics
Place of PublicationPoland
PublisherBanach Center Publications
Publication statusE-pub ahead of print - 1 Sep 2020