rt.representation theory - An identity for sheaf cohomology of flag varieties

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $phi:Grightarrow G/B$ 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

$$Gtimes_B V :=(Gtimes V)/{(g,v)sim(gb^{-1},theta(b)v),forall bin B}.$$
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 $thetaotimeschi_lambda$ 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 $Xtimes V$ with $(g,(x,v))mapsto (gx,pi(g)v)$). Is the identity

$$mathrm{H}^i(X,mathcal{I}(thetaotimeschi_lambda))simeq mathrm{H}^i(X,mathcal{I}(chi_lambda)) otimes V$$
as $G$ or $mathfrak{g}$-modules correct?

Background/Motivation

I have been reading about the Borel-Weil theorem lately, and

Edit:
This question arose from an attempt to fix a mistake in a book.
The identity is indeed correct and I believe I have found the error elsewhere.
Thanks Chuck and Jim!

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