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Let $\g$ be any simple Lie algebra over $\mathbb{C}$. Recall that thereexists an embedding of $\mathfrak{sl}_2$ into $\g$, called a principal TDS,passing through a principal nilpotent element of $\g$ and uniquely determinedup to conjugation. Moreover, $\wedge (\g^*)^\g$ is freely generated (in thesuper-graded sense) by primitive elements $\omega_1, \dots, \omega_\ell$, where$\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitiveelement $\omega \in \wedge^d (\g^*)^\g$, 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 thevalidity of this conjecture for simple simply-laced Lie algebras implies itsvalidity for any simple Lie algebra.  Let G be a connected, simply-connected, simple, simply-laced algebraic groupand let $\sigma$ be a diagram automorphism of G with fixed subgroup K. Then, weshow that the restriction map R(G) \to R(K) is surjective, where R denotes therepresentation ring over $\mathbb{Z}$. As a corollary, we show that therestriction map in the singular cohomology H^*(G)\to H^*(K) is surjective. Ourproof of the reduction of Hitchin's conjecture to the simply-laced case relieson this cohomological surjectivity.

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