Canonical bases of quantum Schubert cells and their symmetries

The goal of this work is to provide an elementary construction of the canonical basis $mathbf B(w)$ in each quantum Schubert cell~$U_q(w)$ and to establish its invariance under modified Lusztigs symmetries. To that effect, we obtain a direct characterization of the upper global basis $mathbf B^{up}$ in terms of a suitable bilinear form and show that $mathbf B(w)$ is contained in $mathbf B^{up}$ and its large part is preserved by modified Lusztigs symmetries.