# 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 $\phi:G\rightarrow G/B$ 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

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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I don't think that's correct; you should have $V$ instead of $V^*$ in the identity. For example, consider the trivial $G$-equivariant bundle on $G/B$ with fiber $V$; its global sections are isomorphic to $V$. – Chuck Hague Sep 4 '10 at 15:00
Silly nitpick: In the second line, $\phi$ should map to $X$, not $B$. – S. Carnahan Sep 5 '10 at 9:46
@Scott: Edited. – Jim Humphreys Sep 10 '10 at 22:36

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.