Let \g be any simple Lie algebra over C. Recall that there
exists an embedding of sl2 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 ω1,…,ωℓ, where
ℓ is the rank of \g. N. Hitchin conjectured that for any primitive
element \omega \in \wedge^d (\g^*)^\g, there exists an irreducible
sl2-submodule V_\omega \subset \g 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) \to 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)\to H^*(K) is surjective. Our
proof of the reduction of Hitchin's conjecture to the simply-laced case relies
on this cohomological surjectivity.Comment: 14 page