Let $G$ be a quasi-split reductive group, over a local field, with a Borel subgroup $B=T\cdot N$ and the associated Weyl group $W$. Given a family of induced representations $\pi\_s = Ind\_B^G \chi\cdot \delta\_B^s$ and an element $w\in W$ one defines an intertwining operator
$$M\_w\left(f\_s\right)\left(t\right) = \int\limits\_{ \left(N\cap w^{-1}Nw \right) \backslash N} f\_s\left(wnt\right) \ dn$$
It is "common knowledge" that given a simple root $\alpha$ of $T$ with respect to $B$ and the corresponding simple reflection $w\_\alpha$, the associated intertwining $M\_{w\_\alpha}$ "behaves" as in the rank-one case. Particularly, in [Ikeda, Tamotsu, On the location of poles of the triple L-functions], Ikeda states (on page 194) that for $G=Sp\_{2n}$ if we denote by $\iota\_\alpha:SL\_2\to G$ be the map corresponding to $\alpha$ then
$$ \iota\_\alpha^\ast\left( M\_{w\_\alpha}\left(f\_s \right) \right) = M\_{w\_0}\iota\_\alpha^\ast \left( f\_s \right) , $$
where $w\_0$ is the non-trivial element in the Weyl group of $SL\_2$ and $\iota\_\alpha^\ast$ is the pullback map.
I am looking for a reference to this statement and what can be said for a gneral quasi-split group.


Link to Ikeda's paper: <http://www.numdam.org/item?id=CM_1992__83_2_187_0>



Intertwining Operators Associated to Simple Reflections
W. Casselman's 1980 paper in Comp. Math. ‘The unramified principal series of p-adic groups, I: the spherical function’, does this.


