Spectral Transfer for Metaplectic Groups. II. Hecke Algebra Correspondences

Abstract

Let $\text {Mp}(2n)$ 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 ($p \ne 2$) and Takeda–Wood ($p=2$) obtained an equivalence between the Bernstein block of $\text {Mp}(2n)$ containing the even (resp. odd) Weil representation and the Iwahori-spherical block of the split $\text {SO}(2n+1)$ (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.

Keywords

Hecke algebra / Metaplectic group / Local Langlands correspondence / Primary 22E50 / Secondary 11F70 / 20C08

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Funding

NSFC(11922101)

RIGHTS & PERMISSIONS

Peking University

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