Irreducibility of the unramified principal series 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.  
 A: Suppose $F$ is a non-archimedean local field. This regularity assumption is not needed, nor is the assumption that $\chi$ is unramified. The sufficient part for irreducibility is part of Theorem 4.2 of Bernstein and Zelevinsky - Induced representations of reductive $\mathfrak p$-adic groups. I and the other part is part of Theorem 6.1 in Zelevinsky - Induced representations of reductive $\mathfrak p$-adic groups. II. On irreducible representations of $\mathrm{GL}(n)$.
Edit: To respond to Vincent's comment, let me say what I know (very limited): Note that when $F$ is a finite field, $\delta$ and $|\cdot|_F$ are trivial and the assertion follows from Mackey theory that
$$\mathrm{Hom}_G(I(\chi),I(\chi))=\bigoplus_{w\in N_G(T)/T}\mathrm{Hom}_T(\chi,{}^w\chi)$$
When one tries to adapt this to (any) local field, the normalization factor $\delta$ and $|\cdot|_F$ should naturally arise. In general $I(\chi)$ is no longer semisimple and one needs some tricks to know a bit more than the $\mathrm{Hom}$. Beyond this I know nothing about how the case $F$ archimedean has in common with the $p$-adic case.
