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We consider reduced imaginary Verma modules for the untwisted quantum affinealgebras $U_q(\hat{\g})$ and define a crystal-like base which we call imaginarycrystal base using the Kashiwara algebra $\mathcal K_q$ constructed in earlierwork by Ben Cox and two of the authors. We prove the existence of the imaginarycrystal base for any object in a suitable category $\mc{O}^q_{red,im}$containing the reduced imaginary Verma modules for $U_q(\hat{\g})$.

Crystal bases for reduced imaginary Verma modules of untwisted quantum affine algebras