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

Abstract

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 $ω_1, \dots, ω_\ell$, where $\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitive element $ω\in \wedge^d (\g^*)^\g$, there exists an irreducible $\mathfrak{sl}_2$-submodule $V_ω\subset \g$ of dimension $d$ such that $ω$ is non-zero on the line $\wedge^d (V_ω)$. 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 $σ$ 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.