A Borel-Weil theorem for the irreducible quantum flag manifolds

Abstract

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 of a number of authors (including the first and third authors of this paper) on the quantum Grassmannians ${\mathcal{O}}_q({\mathrm{Gr}}_{n,m})$. 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 ${\mathcal{O}}_q(G/L_S^{\mathrm{s}})$.

26 pages