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

TL;DR

This paper proves that Hitchin's conjecture for simply-laced Lie algebras implies its validity for all simple Lie algebras, using cohomological methods and properties of representation rings.

Contribution

It establishes a reduction of Hitchin's conjecture from general simple Lie algebras to the simply-laced case via cohomological surjectivity.

Findings

Proof that Hitchin's conjecture for simply-laced Lie algebras implies it for all simple Lie algebras.

Surjectivity of the restriction map in representation rings for automorphisms of algebraic groups.

Surjectivity of the restriction map in singular cohomology from G to K.

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 $\omega_1, \dots, \omega_\ell$, where $\ell$ is the rank of $\g$. N. Hitchin conjectured that for any primitive element $\omega \in \wedge^d (\g^*)^\g$, 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 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 $\sigma$ 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.