Archimedean theory and $\epsilon$-factors for the Asai Rankin-Selberg integrals

TL;DR

This paper advances the local theory of Asai L-functions and epsilon-factors by establishing key functional equations at Archimedean places and proving their equality across all local fields, with implications for representation theory.

Contribution

It completes the local Rankin-Selberg theory for Asai L-functions and epsilon-factors, establishing functional equations and equality of epsilon-factors uniformly across all characteristic zero local fields.

Findings

Established local functional equations at Archimedean places.

Proved equality of Rankin-Selberg and Langlands-Shahidi epsilon-factors at all places.

Unified approach applicable to any characteristic zero local field.

Abstract

In this paper, we partially complete the local Rankin-Selberg theory of Asai $L$-functions and $\epsilon$-factors as introduced by Flicker and Kable. In particular, we establish the relevant local functional equation at Archimedean places and prove the equality between Rankin-Selberg's and Langlands-Shahidi's $\epsilon$-factors at every place. Our proofs work uniformly for any characteristic zero local field and use as only input the global functional equation and a globalization result for a dense subset of tempered representations that we infer from work of Finis-Lapid-M\"uller. These results are used in another paper by the author to establish an explicit Plancherel decomposition for $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$, $E/F$ a quadratic extension of local fields, with applications to the Ichino-Ikeda and formal degree conjecture for unitary groups.