Imaginary Verma modules and Kashiwara algebras for Uq(gˆ)

We consider imaginary Verma modules for quantum affine algebra Uq(gˆ), where gˆ has Coxeter–Dynkin diagram of ADE type, and construct Kashiwara type operators and the Kashiwara algebra KqKq. We show that a certain quotient Nq− of Uq(gˆ) is a simple KqKq-module.