  The pro-$p$-Iwahori Hecke algebra has an involution $\iota$ defined in terms
of Iwahori-Matsumoto basis. Then for a module $\pi$ of pro-$p$-Iwahori Hecke,
$\pi^\iota = \pi\circ \iota$ is also a module. We calculate $\pi^\iota$ for
simple modules $\pi$. We also calculate the dual of $\pi$.
