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](http://www.math.tifr.res.in/~dprasad/ictp2.pdf) on representation theory for $\operatorname{GL}\_n$, but I wasn't sure if they were making an underlying assumption of regularity. 



Irreducibility of the unramified principal series
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](http://www.math.tau.ac.il/~bernstei/Publication_list/publication_texts/BZ-induced-1.pdf) 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 $\mathrm{GL}(n)$](http://www.numdam.org/article/ASENS_1980_4_13_2_165_0.pdf).


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.


