# Adjacent parabolic subgroups and proportionality to $\alpha^{\vee}$

Let $$P = MN$$ be a parabolic subgroup of a $$p$$-adic reductive group $$G$$ with split component $$A_M$$. There is bijection from the set of parabolic subgroups of $$G$$ with Levi $$M$$ and the chambers of $$\mathfrak a_M$$ with respect to the hyperplanes $$H_{\alpha} = \{ h \in \mathfrak a_M : \alpha \in \Phi(A_M,G) \}$$, where $$P$$ corresponds to the chamber

$$\{ h \in \mathfrak a_M : \langle h,\alpha \rangle > 0 \textrm{ for all } \alpha \in \Phi(A_M,N)\}$$

A parabolic subgroup $$P' = MN'$$ is said to be adjacent to $$P$$ if the chambers of $$P$$ and $$P'$$ are separated by a single hyperplane $$H_{\alpha}$$. Then there is a unique simple root $$\alpha$$ of $$A_M$$ in $$N$$ for which $$-\alpha$$ is a simple root of $$A_M$$ in $$N'$$.

Let $$K$$ be a maximal compact open subgroup of $$G$$ in good position relative to $$P$$ and $$P'$$, so that the Harish-Chandra map $$H_P$$ extends to $$G$$ via the equation $$G = PK$$. Let $$\alpha^{\vee} \in \mathfrak a_M$$ be the coroot corresponding to $$\alpha$$, and suppose $$n' \in \overline{N} \cap N'$$. Is it true that $$H_P(n')$$ is proportional to $$\alpha^{\vee}$$? This is claimed in Waldspurger's writeup on Harish-Chandra's notes on the proof of the convergence of intertwining operators:

To compute $$H_P(n')$$, we need to write $$n' = nmk$$ for $$n \in N, m \in M, k \in K$$, so that $$H_P(n') = H_P(m)$$. As a start, I could try to show that $$\langle \beta, H_P(m)\rangle$$ for all $$\beta \in \Delta(A_M,N)$$ except for $$\alpha$$. But I don't see how to use the fact that $$n' \in \overline{N} \cap N'$$ to tell anything about $$m$$.