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We define a new correspondence for pairs $(\mathbb{F},H)$ formed by a flagmanifold $\mathbb{F}$ together with an $H$-flux on $\mathbb{F}$. Given its rolewithin our correspondence, infinitesimal $T$-duality may be viewed as a sourceof $H$-flux, in the sense that it contributes towards taking fluxless pairs$(\mathbb{F},0)$ to pairs $(\mathbb{F}^\vee, H^\vee)$ carrying nontrivial flux$H^\vee\neq 0$. We also illustrate how our correspondence exchanges complexstructures with symplectic ones up to $B$-transformations.

The $H$-flux on flag manifolds generated by infinitesimal $T$-duality