Let $\Phi$ be an irreducible root system of rank $\ell$. The fundamental invariants of $\Phi$ is a set of $\ell$ integers $d_1, \cdots, d_\ell$ canonically attached to $\Phi$.

Now suppose $\Psi$ is a closed subsystem of $\Phi$ and suppose $\Psi$ is generated by a subset of simple roots of $\Phi$. (In the language of Lie algebras, $\Psi$ defines a Levi subalgebra of the simple Lie algebra corresponding to $\Phi$.)

Question 1: Is it true that every fundamental invariant of $\Psi$ is a divisor of a fundamental invariant of $\Phi$?

Now suppose I drop the condition that $\Psi$ is generated by a subset of simple roots of $\Phi$. For example, $G_2$ contains $A_2$ as a closed subsystem, but this subsystem is not generated by simple roots of $G_2$. (In the language of Lie algebra, this is saying that $\Psi$ defines a generalized Levi subalgebra. These arise in the study of parahoric subgroups of the loop group.)

Question 2: What is the relationship between fundamental invariants of $\Psi$ and $\Phi$?