TITLE: Subgroup of $\mathrm{GL}_n$ stabilizing linear subspace skew-symmetric matrices
QUESTION [9 upvotes]: I am currently reading "Schiffer variations and the generic Torelli theorem for hypersurfaces" by Voisin, where it is claimed that the subgroup of $\mathrm{SL}_{2m}$ ($m \geq 3$) which preserves a generic subspace of $\bigwedge^2 \mathbb{C}^{2m}$ of dimension bigger than $3$ must be finite. There are no references given in the paper, where it is claimed that this fact is easily checked. Unfortunately (for me), I am not able to prove it.
Here "preserves" means that $g\cdot w\cdot{\vphantom g^tg} \in W$, for any $w \in W$. Is this fact indeed well-known? I am looking for a reference of this fact, or a quick proof.

REPLY [5 votes]: In the following, a linear group is a (closed) subgroup $G$ of some $GL(V)$. Now the stabilizer of a generic $d$-space in $V$ is the same as the stabilizer in $G\times GL(d)$ of a generic vector of $V\otimes\mathbb C^d$. Thus, in our case we have to investigate the generic stabilizer of $GL(n)\times GL(d)$ acting on $\wedge^2\mathbb C^n\otimes\mathbb C^ d$ with $1\le d\le n(n-1)/4$ and determine when it is infinite.
Determining the generic isotropy group $H$ for a linear action of a reductive group $G$ is a classical problem of invariant theory. It was pretty much settled by the Vinberg school approx. 50 years ago. More specifically: In
Andreev, E. M.; Vinberg, È. B.; Èlašvili, A. G.: Orbits of highest dimension of semisimple linear Lie groups. (Russian) Funkcional. Anal. i Priložen. 1 1967 no. 4, 3–7
the authors derive a numerical criterion for $H$ to be infinite (the first theorem of that paper). The argument is really ingeneous and I recommend reading the paper. From their criterion they derive the classification of irreducible simple linear groups with infinite $H$. The table became later notorious because the very same table showed up in various different classification projects (see, e.g., the appendix to Mumford-Fogarty(-Kirwan)).
Probably, the numerical criterion alone is already enough to settle the finiteness of $H$ in the case at hand. But that is not necessary. In the follow-up paper
Èlašvili, A. G. Stationary subalgebras of points of general position for irreducible linear Lie groups. (Russian) Funkcional. Anal. i Priložen. 6 (1972), no. 2, 65–78
Èlašvili (almost, see below) settles the case of arbitrary irreducible linear groups. The answer for non-simple groups is given in Tables 5 and 6. The cases $\wedge^2\mathbb C^n\otimes\mathbb C^ d$ with infinite $H$ are $(n,d)=(n,1), (n,2), (4,3), (5,3)$, and $(6,3)$. So $d\ge4$ does not occur, as claimed.
There is more to say. Sifting through the tables one will notece that Robert Bryant's case $(n,d)=(6,3)$ is missing in Table 5. This case along with three more missing cases was pointed out in
Popov, A. M. Irreducible semisimple linear Lie groups with finite stationary subgroups of general position. (Russian) Funktsional. Anal. i Prilozhen. 12 (1978), no. 2, 91–92.
See the second paragraph. As for reliability of these results, the whole classification was recovered in
Knop, Friedrich; Littelmann, Peter Der Grad erzeugender Funktionen von Invariantenringen. (German) [The degree of generating functions of rings of invariants] Math. Z. 196 (1987), no. 2, 211–229
There we dealt with a slightly broader classification problem where we nevertheless had to do all calculations from scratch. So we were not using Èlašvili's tables directly. That's why I am pretty positive that the tables are complete.