Suppose $G$ is an algebraic group, with maximal rational torus $T$, corresponding character group $X(T)$, and Weyl group $W$. Then $W$ acts on $X(T)$ via $(w\cdot \chi)(t)=\chi(t^w)$, where $t^w=w^{-1}tw$.
If $\chi_1$ and $\chi_2$ are irreducible and conjugate under $W$ in $X(T)$, and $R_T^G$ denotes the parabolic induction functor, is it true that $R_T^G(\chi_1)$ and $R_T^G(\chi_2)$ are isomorphic as modules? (I'm blurring the distinction between characters and representations.)