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

Abstract

We prove Hitchin's conjecture for $\mathfrak{sl}_n$ in every odd prime degree. More precisely, let $p=2k+1$ be prime and let $n\geq k+1$. Every nonzero primitive element $\alpha\in\Lambda^p(\mathfrak{sl}_n^*)^{\mathfrak{sl}_n}$ is nonzero on the top exterior power of the unique $p$-dimensional irreducible submodule of $\mathfrak{sl}_n$ for the adjoint action of a principal $\mathfrak{sl}_2$-subalgebra. The conjecture also holds for types $B_\ell$ and $C_\ell$, with $\ell\geq2$, in every prime degree $p=4r-1$ with $1\leq r\leq\ell$.