Involutions on pro-$p$-Iwahori Hecke algebras

The pro-$p$-Iwahori Hecke algebra has an involution $\iota$ defined in terms of the 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 $\pi$. These calculations will be used for calculating the extensions between simple modules.