Let $G$ be a reductive algebraic group over the field $\mathbb C$, $B\subset G$ a Borel subgroup and $G/B$ their quotient variety.
For every $B$-module $(M,\rho)$ we can construct the associated $G$-equivariant vector bundle $G\times^BM\to G/B$, where $G\times^BM$ is the quotient of $G\times M$ by the relation $(g,m)\sim (gh,h^{-1}\,m)$.
In quite a few papers I read, the tangent bundle $T(G/B)$ is identified with the associated bundle $G\times ^B T_{eB}(G/B)$, where $B$ acts on $T_{eB}(G/B)$ via the adjoint representation. Why do we have this identification?
I tried to split this problem into two different steps:
Proof that every homogeneous vector bundle over $G/B$ is of the form $G\times ^BM$ for some $H$-module $M$.
Show that $T(G/B)$ is homogeneous and that the representation descends to the adjoint representaiton on the tangent space $T_{eB}(G/B)$.
For the first step let $\pi:E\to G/B$ be a homogeneous vector bundle. Since $B$ acts trivially on $G/B$, it stablizes $M= \pi^{-1}(eB)$, so we can regard $M$ as a $B$-module. Now $[(g,m)]\mapsto g\, m$ induces a $B$-equivariant morphsm of vector bundles $G\times^B M\to E$ over $G/B$, which is by definition an isomorphism of the fibers.
Is this enough to ensure that those bundles are isomorphic, like it is e.g. in topology?
For the second step I found in Gagliardi and Hofscheiers paper "Gorenstein spherical fano varieties" a $G$-linearization on the tangent sheaf, which corresponds to a equivariant vector bundle, but since I am quit new to algebraic geometry as a whole, I dont know how to check, if it induces the adjoint action on the tangent space:
The pullback of differential forms with respect to the action morphism $\alpha:G\times G/B\to G/B$, $g\cdot xB=(gx)B$ yields the inverse of a $G$-linearization of the cotangent sheaf, namely $\hat\alpha^{-1}:\alpha^*\Omega_{G/B}\to \pi^*_{G/B}\Omega_{G/B}$. As $G/B$ is smooth, we may dualize $\hat\alpha^{-1}$ and obtain a $G$-linearization of the tangent sheaf $\mathcal T_{G/B}$, namely $\hat\beta=(\hat \alpha^{-1})^\vee:\pi^*\mathcal T_{G/B}\to \alpha^*\mathcal T_{G/B}.$
For $U\subset G/B$ open affine, $g\in G$ acts on a local section $\delta\in \mathrm{Der}_\mathbb C (\mathbb C[U],\mathbb C[U])=\Gamma(U,\mathcal T_{G/B})$ by $$g\cdot\delta=\hat\beta|_{g\times G/B}(\delta)=\lambda^\#_g\circ \delta\circ \lambda^\#_{g^{-1}}\in \mathrm{Der}_\mathbb C (\mathbb C[g\cdot U],\mathbb C[g\cdot U]),$$where $\lambda_g:G/B\to G/B$ is given by $x\mapsto g\cdot x$.