We construct quantum deformations of imaginary Verma modules over $U(A_1^{(1)})$ and show that, for generic $q$, imaginary Verma modules over $U(A_1^{(1)})$ can be deformed to those over the quantum group $U_q(A_1^{(1)})$ in such a way that the dimensions of the weight spaces are invariant under the deformation. We also prove the PBW theorem for $U_q(A_1^{(1)})$ with respect to the triangular
decomposition induced from the root partition corresponding to the
imaginary Verma modules.
1991 Mathematics Subject Classification: 17B67, 17B65, 17B10.