Pro-\(p\)-Iwahori invariants for \(\mathrm{SL}_{2}\) and \(L\)-packets of Hecke modules (Q2797853)

Let \(p\) be a prime number. Let \(F\) be a non-Archimedean local field of residual characteristic \(p\). This paper concerns transferring results on representations of the tame (or ``pro-\(p\)-Iwahori-Hecke'') algebra (see [the reviewer, J. Lie Theory 21, No. 2, 469--489 (2011; Zbl 1268.22016)]) of the group \(\mathrm{GL}(n,F)\), to representations of the tame algebra of its derived subgroup, \(\mathrm{SL}(n,F)\). Two interesting main results are deduced. The first result is an equivalence of categories between the category of Hecke modules in characteristic \(p\) over the tame algebra of \(\mathrm{SL}(2,\mathbb{Q}_p)\), and the category of smooth mod-\(p\) representations of \(\mathrm{SL}(2,\mathbb{Q}_p)\) which are generated by their tame subgroup (or pro-\(p\)-Iwahori) invariants. The second result is a ``numerical correspondence'' between packets of supersingular Hecke modules in characteristic \(p\) over the tame algebra of \(\mathrm{SL}(n,F)\), and irreducible, \(n\)-dimensional projective Galois representations. These results use and extend those of \textit{E. Grosse-Klönne} [Duke Math. J. 165, No. 8, 1529--1595 (2016; Zbl 1364.11103)].