Composition series of tensor product (Q1940452)

Let \(U_q(\mathfrak{g})\) be the quantized enveloping algebra associated to an arbitrary symmetrizable Kac-Moody algebra \(\mathfrak{g}\), and for a dominant integral weight \(\lambda\) denote by \(V(\lambda)\) (resp. \(V(-\lambda)\)) the corresponding simple highest (resp. lowest) weight \(U_q(\mathfrak{g})\)-module. In the article under review, the authors prove for any pair of dominant integral weights \((\lambda, \mu)\) that there is a composition series of \(V(\lambda) \otimes V(\mu)\) compatible with the canonical basis, that is, consisting of submodules spanned by subsets of the canonical basis. This is a generalization of Lusztig's conjecture which asserts that \(V(\lambda) \otimes V(-\mu)\) has such a composition series when \(\mathfrak{g}\) is of finite type. (Note that in this case \(V(-\mu)\) is also a highest weight module.) In the article, they also prove that there is a composition series of \(V(\lambda) \otimes V(-\mu)\) compatible with canonical basis when \(\mathfrak{g}\) is of affine type and the level of \(\lambda - \mu\) is nonzero.