Question about a notation.

Question

I have a question about the notation in the paper. On page 8, the 3-rd line from bottom, it is said that $h_J(t)$ is the corresponding diagonal matrix in $GL_4$. I think that $J$ is some set such that where $\alpha_J=\sum_{j \in J} \alpha_j$ is a root. For example, let $h_J(t)$ in $GL_6$ and $J=\{3,4,5\}$. What is $h_J(t)$? Why $\alpha_J(h_J(t))=t^2$? Thank you very much.