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 $n\ge k+1$. Every nonzero primitive element $\alpha \in {\Lambda }^p({\mathfrak{sl}}_n^{\ast }{)}^{{\mathfrak{sl}}_n}$ 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 $\ell \ge 2$, in every prime degree $p=4r-1$ with $1\le r\le \ell$.

7 pages