The pro-$p$-Iwahori Hecke algebra has an involution $iota$ defined in termsof Iwahori-Matsumoto basis. Then for a module $pi$ of pro-$p$-Iwahori Hecke,$pi^iota = picirc iota$ is also a module. We calculate $pi^iota$ forsimple modules $pi$. We also calculate the dual of $pi$.
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机译：Pro-$ P $ -iwahori HECKE代数参与IWAHORI-MATSUMOTO的涉及$ iota $。然后对于一个模块$ pi $ pro-$ p $ -iwahori hecke，$ pi ^ iota = pi cirt iota $它也是一个模块。我们计算$ pi ^ iota $ fumple modules $ pi $。我们还计算$ pi $的双重。
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