The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of projective space. See N. Chriss & V. Ginzburg 's book "Representation Theory and Complex Geometry" section 5.7.

Now let $G$ be a simple algebraic group over $\mathbb{C}$, $B$ and $T$ be Borel subgroup and maximal torus contains in $B$, respectively. Let $\mathcal{B}=G/B$ be completed flag variety. Let $\mathcal{O}(\alpha)$ be line bundle over $\mathcal{B}$ parameterized by integral weight($\alpha\in Hom(T,\mathbb{C}^*)$). Also denote it by $\Delta$ the diagonal embedding of $\mathcal{B}$ to $\mathcal{B}\times\mathcal{B}$. My question is: how to give a locally free sheaves resolution of the sheaf $\Delta_*\mathbb{O}(\alpha)$? I hope there is a similiar resolution like Beilinson resolution. Any suggestion is welcome ,thanks!

The Beilinson resolution is a locally free sheaves resolution for sheaf $\Delta_*\mathcal{O}_{\mathbb{P}}$,where $\Delta: \mathbb{P}\to \mathbb{P}\times\mathbb{P}$ is the diagonal embedding of projective space. See N. Chriss & V. Ginzburg 's book "Representation Theory and Complex Geometry" section 5.7.

Now let $G$ be a simple algebraic group over $\mathbb{C}$, $B$ and $T$ be Borel subgroup and maximal torus contains in $B$, respectively. Let $\mathcal{B}=G/B$ be completed flag variety. Let $\mathcal{O}(\alpha)$ be line bundle over $\mathcal{B}$ parameterized by integral weight($\alpha\in Hom(T,\mathbb{C}^*)$). Also denote it by $\Delta$ the diagonal embedding of $\mathcal{B}$ to $\mathcal{B}\times\mathcal{B}$. My question is: how to give a locally free sheaves resolution of the sheaf $\Delta_*\mathcal{O}(\alpha)$? I hope there is a similiar resolution like Beilinson resolution. Any suggestion is welcome ,thanks!