Spectral transfer for metaplectic groups. II. Hecke algebra correspondences

TL;DR

This paper explores the relationship between metaplectic groups and special orthogonal groups through Hecke algebra correspondences, emphasizing the preservation of L-parameters and the role of local root numbers in character differences.

Contribution

It revisits existing Hecke algebra equivalences from an endoscopic perspective, highlighting how L-parameters are preserved and character differences relate to symplectic local root numbers.

Findings

L-parameters of irreducible representations are preserved.

Character differences are governed by symplectic local root numbers.

Endoscopic perspective clarifies the structure of Hecke algebra correspondences.

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.