Tensor products of infinite dimensional modules in the BGG category of a quantized simple Lie algebra of type ADE

We consider the BGG category $\O$ of a quantized universal enveloping algebra $U_q(\mathfrak{g})$. It is well-known that $M\otimes N\in \O$ if $M$ or $N$ is finite dimensional. When $\mathfrak{g}$ is simple and of type ADE, we prove in this paper that $M\otimes N\notin \O$ if $M$ and $N$ are both infinite dimensional.