Simple type theory for metaplectic covers of $\mathrm{GL}(r)$ over a non-archimedean local field

Let $F$ be a non-archimedean locally compact field of residual characteristic $p$, let $G=\mathrm{GL}_{r}(F)$ and let $\widetilde{G}$ be an $n$-fold metaplectic cover of $G$ with $\mathrm{gcd}(n,p)=1$. We study the category $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$ of complex smooth representations of $\widetilde{G}$ having inertial equivalence class $\mathfrak{s}=(\widetilde{M},\mathcal{O})$, which is a block of the category $\mathrm{Rep}(\widetilde{G})$, following the "type theoretical" strategy of Bushnell-Kutzko. Precisely, first we construct a "maximal simple type" $(\widetilde{J_{M}},\widetilde{\lambda}_{M})$ of $\widetilde{M}$ as an $\mathfrak{s}_{M}$-type, where $\mathfrak{s}_{M}=(\widetilde{M},\mathcal{O})$ is the related cuspidal inertial equivalence class of $\widetilde{M}$. Along the way, we prove the forklore conjecture that every cuspidal representation of $\widetilde{M}$ could be constructed explicitly by a compact induction. Secondly, we construct "simple types" $(\widetilde{J},\widetilde{\lambda})$ of $\widetilde{G}$, and prove that each of them is an $\mathfrak{s}$-type of a certain block $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$. When $\widetilde{G}$ is either a Kazhdan-Patterson cover or Savin's cover, the corresponding blocks turn out to be those containing discrete series representations of $\widetilde{G}$. Finally, for a simple type $(\widetilde{J},\widetilde{\lambda})$ of $\widetilde{G}$ we describe the related Hecke algebra $\mathcal{H}(\widetilde{G},\widetilde{\lambda})$, which turns out to be not far from an affine Hecke algebra of type A, and is exactly so if $\widetilde{G}$ is one of the two special covers mentioned above. We leave the construction of a "semi-simple type" related to a general block $\mathrm{Rep}_{\mathfrak{s}}(\widetilde{G})$ to a future phase of the work.