Question

(I apologize in advance for the vagueness of my question). Let $G$ be a reductive algebraic group over $\mathbb C$ with Lie algebra $\frak g$ and Borel $B$. I have seen casual references to the fact that relative Lie algebra cohomology (sometimes also called cohomology of a pair) with coefficients in a $B$-module $V$ is, with the right choice of pair of Lie algebras, related to sheaf cohomology of $G/B$ with coefficients in $V$. Is there a reference for this fact, and is there a precise statement one can make? In particular, which Lie algebras should one use? (My guess would be the pair $(\frak b, \frak h)$, where $\frak b := $ Lie($B$) and $\frak h$ is a Cartan of $\frak b$).

Answer 1

The statement you're looking for is that for a representation $V$ of $G$, the sheaf cohomology of the vector bundle $G\times_B V$ on $G/B$ coincides with the Lie algebra cohomology for $\mathfrak{b}$ acting on $V$. This goes back to Kostant.

Answer 2

As Ben points out, Kostant's papers are a fundamental reference for transition between Bott's work (Annals of Mathematics 66, 1957) involving the flag variety and a more algebraic formulation involving Lie algebra cohomology for the nilradical $\mathfrak{n}$ of a Borel subalgebra. I think the rough intuition here is that for a suitable assignment of positive or negative roots, $\mathfrak{n}$ approximates the flag variety in the classical finite dimensional representation theory setting.

Yet another viewpoint was offered in the 1970s by Bernstein-Gelfand-Gelfand in the context of category $\mathcal{O}$. (For a fairly short treatment of some of these connections see Chapter 6 of my 2008 AMS book GSM 94 where Delorme's formulation in terms of relative Lie algebra cohomology is outlined.) The relative Lie algebra technology was further developed by Borel and Wallach in their monograph (AMS, 2000).

P.S. To clarify the "relative" aspect of the cohomology here, my understanding (probably incomplete) is that in the narrow setting of finite dimensional representations of a semisimple $\mathfrak{g}$, the essential relative Lie algebra cohomology computations involve the pair $(\mathfrak{g},\mathfrak{h})$; here $\mathfrak{h}$ is a Cartan subalgebra lying in $\mathfrak{b}$. But in more sophisticated study of unitary representations of a corresponding noncompact Lie group, with a maximal compact subgroup $K$, the appropriate relative cohomology arises from Harish-Chandra's $(\mathfrak{g}, K)$-modules.