Let $p$ be a prime integer, let $F$ be a non-Archimedean local field of residual characteristic $p$, such as the field $\mathbb{Q}_{p}$ of $p$-adic numbers and let $C$ be an algebraically closed field. In the second half of the past century, Hecke algebras of metaplectic groups and their representation theory have been studied in the setting of Shimura and Theta correspondences. When $C$ is the field of complex numbers, a lot is known for $\widetilde{\mathrm{SL}}_{2}(F)$, but things get much more mysterious when $C$ is of positive characteristic $\ell >0$. Assuming $\ell \not= p$ allows to transfer some of the complex statements to the $\ell$-modular framework, but all collapses when $\ell = p$. In this case (called the $p$-modular case), very little is known about the Hecke algebras associated with $\widetilde{\mathrm{SL}}_{2}(F)$, and basically nothing was done so far regarding the correspondences aforementioned.

In this talk, I will discuss some joint work in progress with Soma Purkait (Tokyo Institute of Technology) in the $p$-modular case. In this setting, we provide a description of the $p$-modular Iwahori-Hecke algebras associated with the $p$-adic metaplectic group $\widetilde{\mathrm{SL}}_{2}(F)$ and of their simple modules. If time allows it, I will also explain how our results compare with what is known for $\mathrm{GL}_{2}(F)$ and $\mathrm{SL}_{2}(F)$, as well as how they connect to the $p$-modular representation theory of $\widetilde{\mathrm{SL}}_{2}(F)$, in the view of a (for now conjectural) $p$-modular Langlands correspondence for this group.

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