Lusztig's positive root vectors and a Dolbeault complex for the A-series full quantum flag manifolds

For the Drinfeld–Jimbo quantum enveloping algebra $U_q\left({\mathrm{sl}}_{n+1}\right)$, we show that the span of Lusztig's positive root vectors, with respect to Littlemann's nice reduced decompositions of the longest element of the Weyl group $S_{n+1}$, form quantum tangent spaces for the full quantum flag manifold $O_q\left(F_{n+1}\right)$. The associated differential calculi are direct q-deformations of the anti-holomorphic Dolbeault complex of the classical full flag manifold $F_{n+1}$. As an application we establish a quantum Borel–Weil theorem for $O_q\left(F_{n+1}\right)$, giving a noncommutative differential geometric realisation of all the finite-dimensional type-1 irreducible representations of $U_q\left({\mathrm{sl}}_{n+1}\right)$. Restricting this differential calculus to the quantum Grassmannians is shown to reproduce the celebrated Heckenberger–Kolb anti-holomorphic Dolbeault complex. Lusztig's positive root vectors for non-nice decompositions of the longest element of $S_{n+1}$ are examined for low orders, and are exhibited to either not give tangents spaces, or to produce differential calculi of non-classical dimension.