An identity for sheaf cohomology of flag varieties

Question

Let $G$ be a connected complex semisimple Lie-group, $T$ a maximal torus and $B$ a Borel subgroup containing it. Let $\phi:G\rightarrow G/B$ 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 $$ G\times_B V :=(G\times V)/\{(g,v)\sim(gb^{-1},\theta(b)v),\forall b\in B\}.$$ 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 $$ \mathrm{H}^i(X,\mathcal{I}(\theta\otimes\chi_\lambda))\simeq \mathrm{H}^i(X,\mathcal{I}(\chi_\lambda)) \otimes V $$ 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!

Answer 1

This may be too naive an answer, but from my experience this kind of question fits comfortably into the foundational material for reductive algebraic groups over an algebraically closed field of arbitrary characteristic. This is for example treated in Part I of J.C. Jantzen's 2003 AMS second edition of Representations of Algebraic Groups. There the starting point is the "tensor identity", followed by "generalized tensor identity" for higher derived functors of induction, which translates for the flag variety into the language of vector bundles and sheaf cohomology. This originates basically in classical Frobenius reciprocity for finite groups but becomes quite flexible in situations involving a reductive group, a Borel (or other parabolic) subgroup, and various finite dimensional rational representations. I'm assuming your V is finite dimensional. As Chuck Hague points out, no dualization should occur in your formula.

Most of what goes into the classical Borel-Weil theory has a natural formulation in any characteristic, though Bott's theorem can't be imitated so precisely for nondominant line bundles.

P.S. There are quite a few literature sources (papers by H.H. Andersen, Cline-Parshall-Scott, Donkin, etc.), but Chapter II.5 in Jantzen's book gives a fairly comprehensive treatment in algebraic language of the theorems of Borel-Weil and Bott, along with a derivation of Weyl's character formula. To get back to the classical theory over $\mathbb{C}$ does require some translation of the language. The elegant papers of Demazure using algebraic geometry in characteristic 0 are the underlying inspiration for much of this approach to ideas first developed in the setting of complex geometry or compact Lie groups.