  Let $\g$ be any simple Lie algebra over $\mathbb{C}$. Recall that there
exists an embedding of $\mathfrak{sl}_2$ into $\g$, called a principal TDS,
passing through a principal nilpotent element of $\g$ and uniquely determined
up to conjugation. Moreover, $\wedge (\g^*)^\g$ is freely generated (in the
super-graded sense) by primitive elements $\omega_1, \dots, \omega_\ell$, where
$\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitive
element $\omega \in \wedge^d (\g^*)^\g$, there exists an irreducible
$\mathfrak{sl}_2$-submodule $V_\omega \subset \g$ 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.
