I'm struggling with something in roots datum theory and representations. This can be summarized in the following two questions (take $G$ a real reductive groupe and $T$ a maximal torus)
1- What are the relationships between the roots datum of $(G,T)$ and $(G^{\text{der}},T^{\text{der}})$?
2-What are the relationships between the representations of $G$ and the representations of $G^{\text{der}}$?