Good grading and isotropic data for affine W-algebra reduction

The nilpotence convention for the supplied $f$ is [F2]. The form ${\omega }_f$ is skew because $[u,v]=-[v,u]$. Invariance gives ${\omega }_f(u,v)=([f,u]\mathrm{\mid }v)$. The good-grading conditions make $\mathrm{ad}f:{\mathfrak{g}}_{1\mathrm{/}2}\to {\mathfrak{g}}_{-1\mathrm{/}2}$ both injective and surjective, hence an isomorphism. Invariance of the form under the grading pairs ${\mathfrak{g}}_{-1\mathrm{/}2}$ nondegenerately with ${\mathfrak{g}}_{1\mathrm{/}2}$, so ${\omega }_f$ is nondegenerate.

All degrees occurring in $\mathfrak{m}$ are positive. Thus the grading rule makes $\mathfrak{m}$ a subalgebra and the invariant form vanish on $\mathfrak{m}\times \mathfrak{m}$, since it pairs only opposite degrees.

A finite-dimensional nondegenerate skew form has a maximal isotropic subspace: if the space is nonzero, choose $u\mathrm{\ne }0$ and then $v$ with ${\omega }_f(u,v)=1$; their span is nondegenerate, its orthogonal complement is nondegenerate of dimension two smaller, and induction supplies a Lagrangian there. Adding $\mathbb{C}u$ gives an isotropic subspace of half the original dimension. The zero-dimensional case uses $\mathrm{\ell }=0$.

The character $\chi$ vanishes on every graded component except degree $1$. In $[\mathfrak{m},\mathfrak{m}]$, only $[\mathrm{\ell },\mathrm{\ell }]\subseteq {\mathfrak{g}}_1$ can contribute to that component, and $\chi ([u,v])={\omega }_f(u,v)=0$ for $u,v\in \mathrm{\ell }$. Hence $\chi ([\mathfrak{m},\mathfrak{m}])=0$, so $\chi$ is a Lie algebra character.