Let $G = \operatorname{GL}_n(F)$ with the usual Borel subgroup $P = TU$. Let $\chi = \chi_1 \otimes \cdots \otimes \chi_n$ be an unramified character of $T$. Suppose that $\chi$ is regular, which is to say that $\chi_i \neq \chi_j$ for $i \neq j$. Equivalently, $\chi \neq w.\chi$ for all $w \in W(T,G)$.

Then $I(\chi) = \operatorname{Ind}_{TU}^G \chi \delta^{1/2}$ is irreducible if and only if $\chi_i \neq \chi_j | \cdot |_F$ for all $i \neq j$. Is this true without the regularity assumption on $\chi$? This is claimed in Example 4.2 of Prasad and Raghuram's notes on representation theory for $\operatorname{GL}_n$, but I wasn't sure if they were making an underlying assumption of regularity.