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.
