The general-position criterion for generalised Gelfand–Graev representations

Let $G$ be a finite reductive group with Frobenius endomorphism $F$, let $k$ be the underlying field of characteristic $p$, and let $d\in\Delta$ index the corresponding grading of the Lie algebra. Write $\mathfrak{g}_d(i)$ for its degree-$i$ component and let $\sigma_\lambda$ be the alternating form on $\mathfrak{g}_d(1)$ associated with a linear map $\lambda\colon\mathfrak{g}_d(2)\to k$. Let $\Delta_k^{\!\bullet}$ denote the set of indices for which the associated form can be non-degenerate. General-position criterion. For every $d\in\Delta$, $d\in\Delta_k^{\!\bullet}$ if and only if either

$$\mathfrak{g}_d(1)=\{0\},$$

or there exists a linear map $\lambda\colon\mathfrak{g}_d(2)\to k$ such that the radical of $\sigma_\lambda$ is zero. This characterises precisely when the corresponding generalised Gelfand–Graev construction admits the required non-degenerate alternating form; the paper presents it as suggested by the preceding examples, and no resolution is supplied.