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

Abstract

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, ∧(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 ω∈∧d(g⁎)g, there exists an irreducible sl2-submodule Vω⊂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)→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.