Canonical bases of tensor products and positivity properties

Let $\mathbf{U}$ be a quantum group of symmetric type. We introduce the {\it thickening realization} to realize (a suitable approximation of) the tensor product ${^{\omega}\Lambda_{\lambda_1}}\otimes \Lambda_{\lambda_2}$ of a simple integrable lowest weight module and a highest weight module as a subquotient of the Verma module of a larger quantum group $\tilde{\mathbf{U}}$. For the canonical basis of the tensor product, we show that the entries of the transition matrix from the pure tensor basis to it, and the structure constants of the action by spherical parabolic subalgebras of the modified quantum group $\dot{\mathbf{U}}$ are given by the structure constants of the comultiplication and multiplication in the negative part $\tilde{\mathbf{U}}^-$ of $\tilde{\mathbf{U}}$ with respect to its canonical basis respectively. Thus, we deduce the positivity property of the canonical basis of the tensor product. In particular, we obtain the positivity property of the canonical bases for the action of $\dot{\mathbf{U}}$ on simple integrable highest weight modules, generalizing Lusztig's theorem from Chevalley generators to any canonical basis elements of $\dot{\mathbf{U}}$; for the action of Chevalley generators on ${^{\omega}\Lambda_{\lambda_1}}\otimes \Lambda_{\lambda_2}$; and for multiplication in $\dot{\mathbf{U}}$, as well as the actions on arbitrary tensor products. At $v=1$, these results connect to geometric total positivity on double flag varieties, explored in the joint work of He and Xie.