Annihilating Fields of Standard Modules for Affine Lie Algebras

Given an affine Kac-Moody Lie algebra $\tilde{\mathfrak{g}}[\sigma]$ of arbitrary type, we determine certain minimal sets of annihilating fields of standard $\tilde{\mathfrak{g}}[\sigma]$-modules. We then use these sets in order to obtain a characterization of standard $\tilde{\mathfrak{g}}[\sigma]$-modules in terms of irreducible loop $\tilde{\mathfrak{g}}[\sigma]$-modules, which proves to be a useful tool for combinatorial constructions of bases for standard $\tilde{\mathfrak{g}}[\sigma]$-modules.