Variation of additive characters in the transfer for Mp(2n)

Let $\mathrm{Mp}(2n)$ be the metaplectic group of rank $n$ over a local field $F$ of characteristic zero. In this note, we determine the behavior of endoscopic transfer for $\mathrm{Mp}(2n)$ under variation of additive characters of $F$. The arguments are based on properties of transfer factor, requiring no deeper results from representation theory. Combined with the endoscopic character relations of Luo, this provides a simple and uniform proof of a theorem of Gan-Savin, which describes how the local Langlands correspondence for $\mathrm{Mp}(2n)$ depends on the additive characters.

💡 Research Summary

The paper studies the dependence of endoscopic transfer and the local Langlands correspondence (LLC) for the metaplectic group Mp(2n) on the choice of additive character ψ of a local field F of characteristic zero. The metaplectic group is introduced as a two‑fold central extension 1 → µ₂ → ˜G^{(2)} → Sp(2n) → 1, which is then pushed out to an eight‑fold cover 1 → µ₈ → ˜G → Sp(2n) → 1. The eight‑fold cover ˜G depends on ψ (through the Schrödinger model of the Weil representation) while the two‑fold cover does not. Genuine representations of ˜G are identified with those of ˜G^{(2)}.

Section 2 reviews the endoscopic data for ˜G. Elliptic endoscopic data are in bijection with pairs (n′, n″) with n′+n″=n; the corresponding endoscopic group is G! = SO(2n′+1) × SO(2n″+1). The transfer factor Δ(γ, \tildeδ) defined in