# Intertwining Operators Associated to Simple Reflections

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

## 1 Answer

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