Action on sheaf cohomology in Bott-Borel-Weil theorem

Question

Let $G$ be simply connected complex semisimple Lie grou and $P \subseteq G$ parabolic subgroup. Suppose $V$ is finite dimensional irreducible representation of $P$ with highest weight $\lambda$, and $G \times _P V$ associated homogeneous vector bundle over $G/P$. Let $\mathcal O _P(\lambda)$ be sheaf holomorphic sections of that bundle.

The Bott-Borel-Weil theorem describes structure of sheaf cohomology $H^r(G/P,\mathcal O _P(\lambda))$ as a $G$-module. (My reference is Baston, Eastwood: The Penrose transform, its interaction with representation theory - chapter 5.)

What I am missing is: How is $G$-action on $H^r(G/P,\mathcal O _P(\lambda))$ apriori defined?

I know that $G$ acts on global sections of $\mathcal O _P(\lambda))$. Is it posible to make some kind of acyclic resolution of $\mathcal O _P(\lambda)$ with $G$-module structure on global sections?

Answer 1

I do not have a complete answer, but it is nevertheless too long for a comment.

If $X$ is a $G$-variety equipped with a $G$-equivariant vector bundle $\xi: E\to X$ (the $G$-equivariant structure in your case is described in Jyrki's comment), then the sheaf of sections $\mathscr F$ of $E$ is a $G$-equivariant quasi-coherent (even locally free) sheaf on $X$ in the sense that, denoting $\rho,\pi: G\times X\to X$ the action resp. projection map, it comes equipped with an isomorphism $\rho^{\ast}{\mathscr F}\cong\pi^{\ast}{\mathscr F}$ that is associative and unital. Denote the category of such sheaves by $\text{QCoh}^G(X)$. In case $X=\text{pt}$, this gives the rational $G$-representations, and any $G$-equivariant morphism $f: X\to Y$ induces a pushforward $\text{QCoh}^G(X)\to\text{QCoh}^G(Y)$ which in case of $Y=\text{pt}$ is the global section functor.

Now two questions have to be addressed:

• Does $\text{QCoh}^G(X)$ have enough injectives?

If yes, one can define the right derived functors $\textbf{R}^{i}f_{\ast}:\text{QCoh}^G(X)\to\text{QCoh}^G(Y)$.

• Does $\text{QCoh}^G(X)\to\text{QCoh}(X)$ preserve injectives, or at least map injectives to $f_{\ast}$-acyclics?

If yes, then the equivariant and non-equivariant versions of $\textbf{R}^if_{\ast}$ are compatible.

In particular, taking $Y=\text{pt}$, the sheaf cohomology of any $G$-equivariant quasi-coherent sheaf admits a canonical structure of a rational $G$-representation, as desired.

Concerning the first question, I found something in Bezrukavnikov, Perverse Coherent Sheaves http://arxiv.org/pdf/math/0005152.pdf:

The forgetful functor $\varepsilon: \text{QCoh}^G(X)\to\text{QCoh}(X)$ admits an exact right adjoint $$\text{av} := \rho_{\ast}\pi^{\ast}:\text{QCoh}(X)\to\text{QCoh}^G(X)$$ which preserves injectives by exactness of $\varepsilon$. Moreover, $\text{av}$ is exact and the unit morphism $\text{id}\to\text{av}\circ\varepsilon$ is a monomorphism, so given any ${\mathscr F}\in\text{QCoh}^G(X)$, an embedding into an injective, $G$-equivariant quasi-coherent sheaf can be constructed as $${\mathscr F}\hookrightarrow\text{av}(\varepsilon {\mathscr F})\hookrightarrow\text{av}({\mathscr I})$$ where $\varepsilon{\mathscr F}\hookrightarrow{\mathscr I}$ is a monomorphism with ${\mathscr I}$ injective in $\text{QCoh}(X)$. Hence $\text{QCoh}^G(X)$ has enough injectives.

At first I was confused about this, since I rather expected a left adjoint of the forgetful functor, but a look at the case $X=\text{pt}$ helps: rational $G$-representations are comodules over the coalgebra $k[G]$, and in the category of comodules over a $k$-coalgebra $C$, the forgetful functor to $k\text{-Mod}$ has the cofree-comodule-functor $-\otimes_k C$ as its right adjoint.

Unfortunately, I cannot say anything about the second question, but maybe someone else does?

Answer 2

For the record, this is a skecth of what I have found out.

Let $\rho \colon G \to G/P$. It is easy to see that $\Gamma(U,\mathcal O_P(\lambda)) \cong \{f \colon \rho^{-1}(U) \to V \text{ such that } f(gp)= p^{-1} \cdot f(g) \text{ for all } g \in G, p \in P\}$.

Take $X \in \mathfrak g$ and consider it as a left invariant vector field on $G$. Then $X$ acts on $\Gamma(U,\mathcal O_P(\lambda))$, and we get an endomorphism of sheaf $\mathcal O_P(\lambda) \to \mathcal O_P(\lambda)$. Since $H^r(G/P,-)$ is a functor, we get an endomorphism $H^r(G/P,\mathcal O_P(\lambda)) \to H^r(G/P,\mathcal O_P(\lambda))$ (by the way, this space is finite dimensional by Cartan-Serre theorem). Since $G$ is simply connected, this action integrates to the action of $G$ on $H^r(G/P,\mathcal O_P(\lambda))$. This construction is from J. L. Taylor: Several complex variables with connections to algebraic geometry and Lie groups.

Alternatives to this and the answer of Hanno are following:

Use isomorphism between sheaf cohomology of $\mathcal O_P(\lambda)$ and Dolbeault cohomology of $G/P$. (R. Zierau: Representations in Dolbeault cohomology; A. W. Knapp: Introduction to representations in analytic cohomology)

Or use Čech cohomology of $\mathcal O_P(\lambda)$, which is, on a paracompact space, same as derived-functor cohomology. (D. N. Akhiezer: Lie group actions in complex analysis)