Hitchin's conjecture for simply-laced Lie algebras implies that for any simple Lie algebra

Let $\text{g}$ be any simple Lie algebra over $\mathbb{C}$. Recall that there exists an embedding of ${\mathfrak{s}\mathfrak{l}}_2$ into $\text{g}$, called a principal TDS, passing through a principal nilpotent element of $\text{g}$ and uniquely determined up to conjugation. Moreover, $\wedge ({\text{g}}^{\ast }{)}^{\text{g}}$ is freely generated (in the super-graded sense) by primitive elements ${\omega }_1,\dots ,{\omega }_{\mathrm{\ell }}$, where $\mathrm{\ell }$ is the rank of $\text{g}$. N. Hitchin conjectured that for any primitive element $\omega \in {\wedge }^d({\text{g}}^{\ast }{)}^{\text{g}}$, there exists an irreducible ${\mathfrak{s}\mathfrak{l}}_2$-submodule $V_{\omega }\subset \text{g}$ of dimension $d$ such that $\omega$ is non-zero on the line ${\wedge }^d(V_{\omega })$. We prove that the validity of this conjecture for simple simply-laced Lie algebras implies its validity for any simple Lie algebra. Let G be a connected, simply-connected, simple, simply-laced algebraic group and let $\sigma$ be a diagram automorphism of G with fixed subgroup K. Then, we show that the restriction map R(G) \to R(K) is surjective, where R denotes the representation ring over $\mathbb{Z}$. As a corollary, we show that the restriction map in the singular cohomology H^(G)\to H^(K) is surjective. Our proof of the reduction of Hitchin’s conjecture to the simply-laced case relies on this cohomological surjectivity.