Spectral transfer for metaplectic groups. II. Hecke algebra correspondences
Fei Chen, Wen-Wei Li
TL;DR
The paper develops an endoscopic perspective on the known Hecke-algebra correspondences between the metaplectic group Mp(2n) and the Iwahori-spherical blocks of SO(2n+1) and its inner form, showing that the Takeda–Wood equivalence TW^* preserves L-parameters while adjusting the component-group characters by symplectic local root numbers. The main quantitative outcome is χ^circ = χ ν_φ, where ν_φ is defined from ε(1/2, φ_i, ψ) for the symplectic-type simple summands φ_i of the L-parameter φ. The authors establish φ^circ = φ and reduce the proof to tempered, good-parity, and square-integrable cases via a sequence of reductions using Jacquet modules, normalized intertwining operators, and endoscopic character relations (Luo). The approach relies on a combination of LLC for the metaplectic and orthogonal sides, compatibility results for TW with parabolic induction and Jacquet functors (CL23), and endoscopic transfers (Luo; Li), to relate transfer of stable distributions and Langlands quotients across groups. The results deepen the connection between endoscopy and Hecke-algebra correspondences for covering groups, and provide a robust local data bridge (via ν_φ) that advances understanding of the metaplectic local intertwining relation and global-to-local applications.
Abstract
Let $\mathrm{Mp}(2n)$ be the metaplectic group over a local field $F \supset \mathbb{Q}_p$ defined by an additive character of $F$ of conductor $4\mathfrak{o}_F$. Gan-Savin ($p \neq 2$) and Takeda-Wood ($p=2$) obtained an equivalence between the Bernstein block of $\mathrm{Mp}(2n)$ containing the even (resp. odd) Weil representation and the Iwahori-spherical block of the split $\mathrm{SO}(2n+1)$ (resp. its non-split inner form), by giving an isomorphism between Hecke algebras. We revisit this equivalence from an endoscopic perspective. It turns out that the L-parameters of irreducible representations are preserved, whilst the difference between characters of component groups is governed by symplectic local root numbers.