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

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Let be any simple Lie algebra over C. Recall that there exists an embedding of sl₂ into, called a principal TDS, passing through a principal nilpotent element of and uniquely determined up to conjugation. Moreover, () is freely generated (in the super-graded sense) by primitive elements ω₁,, ωℓ, where ℓ is the rank of. N. Hitchin conjectured that for any primitive element ω∈ ᵈ (), there exists an irreducible sl₂-submodule Vω⊂ of dimension d such that ω is non-zero on the line ᵈ (Vω). 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) → R(K) is surjective, where R denotes the representation ring over Z. As a corollary, we show that the restriction map in the singular cohomology H*(G)→ H*(K) is surjective. Our proof of the reduction of Hitchin's conjecture to the simply-laced case relies on this cohomological surjectivity.