Pardon if this is well known. Suppose I have a (say complex) connected reductive group $G^{\vee}$ with the $\tilde{\Delta}=\Delta\cup\{\alpha\_0\}$ being the simple roots plus the negative highest root of $G^{\vee}$. Any proper subset of $\tilde{\Delta}$ defines a connected reductive subgroup $H^{\vee}\subset G^{\vee}$. Let $H$ and $G$ be the corresponding dual groups. 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 $H$?
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Thanks!



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> **Edited:** Just to say what I understand / guess so far: For any $u\_H$, say the Springer fiber $\mathcal{B}\_{e\_H}$ at $u\_H$ has dimension $e\_H$. Then the corresponding Weyl group representation appears as the component of $H^{2e\_H}(\mathcal{B}\_{e\_H})$ on which $A\_{e\_H}$ acts trivially. This component receives a surjection from $H^{2e\_H}(H/B\_H)$, which is a quotient of $\text{Sym}^{2e\_H}(std\_H)$, where $std\_H$ is the standard representation of $W\_H$. The Lusztig-Spaltenstein $j$-induction then, I think (I cannot find a reference for why this works for $H^{2e\_H}(H/B\_H)$ instead of $\text{Sym}^{2e\_H}(std\_H)$), gives a $W$-representation in $H^{2e\_H}(G/B)$, which then corresponds to some orbit in $G$ whose Springer fiber has dimension $e\_H$.
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Equality of codimension under Lusztig-Spaltenstein induction
What you're asking for can be phrased entirely in terms of nilpotent orbits. If $\mathcal{O}$ is the nilpotent orbit of $H$ then the nilpotent orbit of $G$ you obtain via your process is the induced nilpotent orbit $\mathrm{Ind}\_H^G(\mathcal{O})$. It is well known that the codimensions of these orbits coincide. See Proposition 7.1.4 of the book "Nilpotent Orbits in Semisimple Lie Algebras" by Collingwood and McGovern.


EDIT: Sorry, I just saw that you want this for a more general class of subgroups. The statement is still true. See 13.3 in Lusztig's book "Characters of reductive Groups over a finite field" and also Lusztig's paper "Unipotent classes and special Weyl group representations", J. Algebra (321), no. 11, 3418-3449, where more details are provided concerning some of the statements in the book.


