2 problems

Speculation on the shifted representation. For every $z\in P$, there is an isomorphism of $\mathcal{H}_I$-modules

Let $F$ be a $p$-adic field, let $G=\mathrm{GL}_n(F)$, and let $\mathcal H$ be the pro-$p$-Iwahori Hecke algebra of $G$. Consider $n$-dimensional modulo $p$ Galois representations…