  We study imaginary representations of the Khovanov-Lauda-Rouquier algebras of
affine Lie type. Irreducible modules for such algebras arise as simple heads of
standard modules. In order to define standard modules one needs to have a
cuspidal system for a fixed convex preorder. A cuspidal system consists of
irreducible cuspidal modules---one for each real positive root for the
corresponding affine root system ${\tt X}_l^{(1)}$, as well as irreducible
imaginary modules---one for each $l$-multipartition. We study imaginary modules
by means of `imaginary Schur-Weyl duality'. We introduce an imaginary analogue
of tensor space and the imaginary Schur algebra. We construct a projective
generator for the imaginary Schur algebra, which yields a Morita equivalence
between the imaginary and the classical Schur algebra. We construct imaginary
analogues of Gelfand-Graev representations, Ringel duality and the Jacobi-Trudy
formula.
