  A cuspidal system for an affine Khovanov-Lauda-Rouquier algerba $R_\al$
yields a theory of standard modules. This allows us to classify the irreducible
modules over $R_\al$ up to the so-called imaginary modules. We make a
conjecture on reductions modulo $p$ of irreducible $R_\al$-modules, which
generalizes James Conjecture. We also describe minuscule imaginary modules,
laying the groundwork for future study of imaginary Schur-Weyl duality. We
introduce colored imaginary tensor spaces and reduce a classification of
imaginary modules to one color. We study the characters of cuspidal modules. We
show that under the Khovanov-Lauda-Rouquier categorification, cuspidal modules
correspond to dual root vectors.
