Composition Series of Tensor Product

Given a quantized enveloping algebra $U_q(\mathfrak g)$ and a pair of dominant weights ($λ$, $μ$), 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 $V(λ)\otimes V(μ)$ 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 $V(λ)\otimes V(-μ)$ compatible with the canonical basis when $\mathfrak g$ is of affine type and the level of $λ-μ$ is nonzero.

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