Cuspidal systems for affine Khovanov-Lauda-Rouquier algebras

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.