Hitchin's Conjecture for ${\mathfrak{s}\mathfrak{l}}_n$ in Odd Prime Degree

We prove Hitchin’s conjecture for ${\mathfrak{s}\mathfrak{l}}_n$ in every odd prime degree. More precisely, let $p=2k+1$ be prime and let $n\ge k+1$. Every nonzero primitive element $\alpha \in {\mathrm{\Lambda }}^p({\mathfrak{s}\mathfrak{l}}_n^{\ast }{)}^{{\mathfrak{s}\mathfrak{l}}_n}$ is nonzero on the top exterior power of the unique $p$-dimensional irreducible submodule of ${\mathfrak{s}\mathfrak{l}}_n$ for the adjoint action of a principal ${\mathfrak{s}\mathfrak{l}}_2$-subalgebra. The conjecture also holds for types $B_{\mathrm{\ell }}$ and $C_{\mathrm{\ell }}$, with $\mathrm{\ell }\ge 2$, in every prime degree $p=4r-1$ with $1\le r\le \mathrm{\ell }$.