Fix number field $F$, its ring of adeles $\mathbb{A}$, a "nice" algebraic group defined over $F$ (at least reductive but for my purposes I can assume simple and simply connected) and a parabolic subgroup (over $F$) $P=M\cdot U$.
Fix an irreducible representation $\sigma$ of $M(\mathbb{A})$ (Either cuspidal or not, I actually care for characters) and consider a subquotient $\Pi$ (a sub, a quotient or a sub of a quotient) of the parabolic induction $Ind\_{P(\mathbb{A})}^{G(\mathbb{A})} \sigma$.
By Flath's theorem, if $\Pi$ is irreducible then it is factorizable in the sense that it can be written as a restirted tensor product $\otimes'\_\nu \Pi\_\nu$.


Here $G$ is defined over the ring of integers a.e. and hence $G(\mathcal{O}\_\nu)$ is defined a.e. and hence also the notion of "spherical vector".


My question is as follows: What if $\Pi$ is not irreducible, will it still be factorizable in a similar sense? If so, is there a reference and if not can somebody suggest a counter example?


Just to make my intentions clearer, I am interested in "special values" of Eisenstein series, namely what is the representation generated by the leading term of Eisenstein series at points of holomorphicity (which won't be part of the discrete spectrum).



Factorizability of Subquotients of Principal Series Representations
It is not even true for finite direct products. Take $G = G\_1 \times G\_2$,
$\chi\_i, \chi\_i'$ to be characters of $G\_i$. Then let $\rho = (\chi\_1 \otimes \chi\_2) \oplus (\chi\_1' \otimes \chi\_2')$. This is 2-dimensional, and the restriction $\rho\_i \simeq \chi\_i \oplus \chi\_i'$ of $\rho$ to each $G\_i$ component is also 2-dimensional. So $\rho$ cannot be a tensor product of $\rho\_1$ with $\rho\_2$.


