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In this paper we view pro-$p$ Iwahori subgroups $I$ as rigid analytic groups$\Bbb{I}$ for large enough $p$. This is done by endowing $I$ with a natural$p$-valuation, and thereby generalizing results of Lazard for $\text{GL}_n$. Wework with a general connected reductive split group over some $p$-adic field(with simply connected derived group) and study the $\Bbb{I}$-analytic vectorsin principal series representations. Our main result is an irreducibilitycriterion which generalizes results of Clozel and Ray in the$\text{GL}_n$-case.

Rigid vectors in $p$-adic principal series representations