I was calculating something with the root system $A_n$ and I think there might be a more general principle at work.
Here is the example: let $G = \operatorname{GL}_5$, with maximal torus $T$ and maximal parabolic $P = MN$, with Levi subgroup $M = \operatorname{GL}_2 \times \operatorname{GL}_3$ and unipotent radical
$$N = \begin{pmatrix} 1 & & \ast & \ast & \ast \\ & 1 & \ast & \ast & \ast \\ & & 1 \\ & & & 1 \\ & & & & 1\end{pmatrix}$$
Let $\rho$ be half the sum of the roots of $T$ in $N$: $\rho = \frac{1}{2}(3e_1 + 3e_2 - 2e_3 - 2 e_4 - 2e_5)$. If $v \in X(T) \otimes \mathbb R$, set $\langle \rho, v \rangle = 2 \frac{(\rho,v)}{(v,v)}$, where $(-,-)$ is the usual inner product. I noticed that if we pair the simple root $e_2-e_3$ defining $P$ with $\rho$, we get something nonzero:
$$\langle \rho,e_2-e_3\rangle = \frac{5}{2}$$
while if we pair any other simple root with $\rho$, we get zero. $\blacksquare$
So my question: let $(V,\Phi)$ be a root system (not necessarily reduced) with base $\Delta$. Fix $\alpha \in \Delta$, and let
$$\Sigma = \{ \gamma = \sum\limits_{\beta \in \Delta} c_{\beta}\beta \in \Phi^+ : c_{\alpha} \neq 0 \}$$
$$\rho = \frac{1}{2}\sum\limits_{\gamma \in \Sigma} \gamma$$
Identify the coroots as elements of the dual $V^{\ast}$. Is it the case that for $\beta \in \Delta$:
$$\beta^{\vee}(\rho) = \begin{cases} 0 & \textrm{ if $\beta \neq \alpha$} \\ \textrm{nonzero} & \textrm{ if $\beta = \alpha$} \end{cases}$$?
