quantum mechanics - Why is $langle hat{J}_x rangle=langle hat{J}_y rangle=0$ if we have a state invariant under rotations about the $z$-axis?

In Leslie E Balletine chapter 8 they state that for a state $\rho=|jm\rangle \langle jm|$ that is an eigenstate of $\mathbf{J}^2$ and $J_z$ we have that $\langle J_x\rangle=\langle J_y\rangle=0$ and $\langle J_x^2\rangle=\langle J_y^2\rangle$ How do you prove this? What are the physical implications?