Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G$ with the $\tilde{\Delta}=\Delta\cup\{-\rho\}$ being the simple roots plus the negative highest root. Any proper subset of $\tilde{\Delta}$ defines a connected reductive subgroup $H\subset G$. For any special nilpotent orbit $u_H$ in $H$ we can define the Lusztig-Spaltenstein induction $u$ as a nilpotent orbit in $G$.

(that is, $(u_H,triv)$ corresponds to some representation of $W_H$ by Springer correspondence, we do the $j$-induction on this to get a representation of $W$, which correspond to $(u,triv)$ for some $u$)

Question: Is it true that the codimension of $u$ in the nilpotent cone of $G$ is the same is the codimension of $u_H$ in the nilpotent cone of $H$?

Thanks!