Composition Series of Tensor Product

Given a quantized enveloping algebra $U_q(\mathfrak{g})$ and a pair of dominant weights ($\lambda$, $\mu$), we extend a conjecture raised by Lusztig in \cite{Lusztig:1992}to a more general form and then prove this extended Lusztig’s conjecture. Namely we prove that for any symmetrizable Kac-Moody algebra $\mathfrak{g}$, there is a composition series of the $U_q(\mathfrak g)$-module$V(\lambda)\otimes V(\mu)$ compatible with the canonical basis. As a byproduct, the celebrated Littlewood-Richardson rule is derived and we also construct, in the same manner, a composition series of $V(\lambda)\otimes V(-\mu)$compatiblewiththecanonicalbasiswhen$\mathfrak g$ is of affine type and the level of $\lambda -\mu$ is nonzero.