Polynomial invariants of Weyl groups for Kac-Moody groups

TL;DR

This paper proves that the polynomial invariants of Weyl groups for certain Kac-Moody Lie algebras are generated by invariant forms, confirming a conjecture and exploring implications for Kac-Moody groups and flag manifolds.

Contribution

It establishes the structure of polynomial invariants for Weyl groups of indecomposable indefinite Kac-Moody Lie algebras, confirming a longstanding conjecture.

Findings

Invariant ring generated by symmetric bilinear form if symmetrizable

Invariant ring is trivial if not symmetrizable

Implications for rational homotopy types of Kac-Moody groups

Abstract

In this paper, we prove that the ring of polynomial invariants of the Weyl group for an indecomposable and indefinite Kac-Moody Lie algebra is generated by invariant symmetric bilinear form or is trivial depending on $A$ is symmetrizable or not. The result was conjectured by Moody and assumed by Kac. As applications we discuss the rational homotopy types of Kac-Moody groups and their flag manifolds.