Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow B$ denote the projection. Given a representation ($\theta,V$) of $B$, we can define a $G$-equivariant holomorphic vector bundle over the flag variety $X:=G/B$ by $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ Its sheaf of sections $\mathcal{I}(\theta)$ may be described as the holomorphic functions $$\mathcal{I}(\theta)(U)=\{f:\phi^{-1}(U)\rightarrow V \mid f(gb^{-1}) = \theta(b)f(g)\} $$ $G$ acts on a section by $(gf)(x)=f(g^{-1}x)$.

An integral weight $\lambda$ of $T$ gives a character $\chi_\lambda$ of $B$. Let $\theta\otimes\chi_\lambda$ denote the tensor product of the representations $\theta$ and $\chi_\lambda$. Suppose ($\theta,V$) is the restriction of a representation ($\pi,V$) of $G$, then the associated ($G$-equivariant) vector bundle is trivial (i.e. isomorphic to $ X\times V$ with $(g,(x,v))\mapsto (gx,\pi(g)v)$). Is the identity $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda)\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V^* $$ (with $V^*$ the dual representation) as $G$ or $\mathfrak{g}$-modules true?

Background: I have been reading about the Borel-Weil theorem lately, and came across this book. Unfortunately, the proof is incorrect (in fact, even the statement of the Borel-Weil(-Bott) theorem is incorrect, as the book uses the above conventions). I'm guessing the mistake is in Lemma 16.4.1, where the above identity would be required with $\lambda=w(\rho)-\rho$, where $\rho$ denotes half the sum of the positive roots and $w$ is some element in the Weyl group. I'm not very familiar with this theory, but hoping that the argument can be fixed.