A Borel–Weil Theorem for the Irreducible Quantum Flag Manifolds.

We establish a noncommutative generalisation of the Borel–Weil theorem for the Heckenberger–Kolb calculi of the irreducible quantum flag manifolds |${\mathcal {O}}_q(G/L_S)$|⁠ , generalising previous work for the quantum Grassmannians |${\mathcal {O}}_q(\textrm {Gr}_{n,m})$|⁠. As a direct consequence we get a novel noncommutative differential geometric presentation of the quantum coordinate rings |$S_q[G/L_S]$| of the irreducible quantum flag manifolds. The proof is formulated in terms of quantum principal bundles, and the recently introduced notion of a principal pair, and uses the Heckenberger and Kolb first-order differential calculus for the quantum Possion homogeneous spaces |${\mathcal {O}}_q(G/L^{\,\textrm {s}}_S)$|⁠. [ABSTRACT FROM AUTHOR]