Explicit determination of the Picard group of moduli spaces of semistable G-bundles on curves

Abstract

Let G be a connected, simply-connected, simple affine algebraic group and Cg be a smooth irreducible projective curve of any genus g ≥ 1 over C. Denote by MCg (G) the moduli space of semistable principal G-bundles on Cg. Let Pic(MCg (G)) be the Picard group of MCg (G) and let X be the infinite Grassmannian of the affine Kac-Moody group associated to G. It is known that Pic(X) Z and is generated by a homogenous line bundle Lχ0 . Also, as proved by KumarNarasimhan [KN], there exists a canonical injective group homomorphism