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For the Drinfeld-Jimbo quantum enveloping algebra $U_q(\frak{sl}_{n+1})$, weshow that the span of Lusztig's positive root vectors, with respect toLittlemann's nice reduced decompositions of the longest element of the Weylgroup, form quantum tangent spaces for the full quantum flag manifold$\mathcal{O}_q(\mathrm{F}_{n+1})$. The associated differential calculi aredirect $q$-deformations of the anti-holomorphic Dolbeault complex of theclassical full flag manifold $\mathrm{F}_{n+1}$. As an application we establisha quantum Borel-Weil theorem for the $A_n$-series full quantum flag manifold,giving a noncommutative differential geometric realisation of all thefinite-dimensional type-$1$ irreducible representations of$U_q(\frak{sl}_{n+1})$. Restricting this differential calculus to the quantumGrassmannians is shown to reproduce the celebrated Heckenberger-Kolbanti-holomorphic Dolbeault complex. Lusztig's positive root vectors fornon-nice decompositions of the longest element of the Weyl group are examinedfor low orders, and are exhibited to either not give tangents spaces, or toproduce differential calculi of non-classical dimension.

Lusztig's Quantum Root Vectors and a Dolbeault Complex for the A-Series Full Quantum Flag Manifolds