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

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