I am reading about some decompositions of spaces of differential forms under the action of the unitary group. The set-up is the following.
Let us take the space of the endomorphisms on a complex vector space $(V,g,J)$. Here I interpret $V$ as a real, $2n$-dimensional vector space equipped with a scalar product $g$ and a complex structure $J$. We know that $\Lambda^2 V = [\![\Lambda^{2,0}]\!] \oplus [\Lambda^{1,1}]$, where $[\![\Lambda^{2,0}]\!]$ is the $(-1)$-eigenspace of $J$ extended to two-forms, whereas $[\![\Lambda^{1,1}]\!]$ is its $(+1)$-eigenspace. Define the two-form $\omega = g(J{}\cdot{},{}\cdot{})$. Since $J\omega = \omega$ we have a further splitting $$\Lambda^2 = [\![\Lambda^{2,0}]\!] \oplus [\Lambda^{1,1}_0] \oplus \mathbb R \omega.$$ This is a unitary decomposition.
My question is the following. The group $\mathrm{U}(n)$ acts on $V$ preserving the whole structure. How can I see that $[\Lambda^{1,1}_0]$ is an irreducible submodule? On the paper I am reading, the authors mention they are using the "Weyl correspondence". When I look up for this I find complicated formulas related to Weyl correspondences of some types, but I doubt they refer to what I need. What is the way to see this point?