Kashiwara Algebras and Imaginary Verma Modules for $U_q(\widehat{\mathfrak{g}})$

We consider imaginary Verma modules for quantum affine algebra $U_q(\widehat{\mathfrak{g}})$, where $\widehat{\mathfrak{g}}$ is of type 1 i.e. of non-twisted type, and construct Kashiwara type operators and the Kashiwara algebra ${\mathcal{K}}_q$. We show that a certain quotient ${\mathcal{N}}_q^{-}$ of $U_q(\widehat{\mathfrak{g}})$ is a simple ${\mathcal{K}}_q$-module.