  Over any non-Archimedean local field of characteristic not equal to $2$,
Takeda and Wood constructed types for the two blocks containing the even and
odd Weil representations of the metaplectic group $\tilde{G}$, and identified
the resulting Hecke algebras $H_\psi^{\pm}$ with the Iwahori-Hecke algebras of
odd orthogonal groups $G^{\pm}$ of the same rank. We describe normalized
parabolic induction and Jacquet modules in terms of Hecke modules using a
suitable variant of Bushnell-Kutzko theory. Furthermore, we match the standard
intertwining operators of $\tilde{G}$ and $G^{\pm}$ by proving a variant of
Gindikin-Karpelevich formula for $\tilde{G}$. As an application, we describe
the behavior of normalized intertwining operators of $\tilde{G}$ in these
blocks under Aubert involution, reducing everything to the $G^{\pm}$ side. This
is mainly motivated by Arthur's local intertwining relations.
