The Borel–Weil–Bott theorem « Concrete Nonsense

Posted by: Steven Sam | April 15, 2009

The Borel–Weil–Bott theorem

In connection with my last post on the Boij–Söderberg conjectures, I mentioned constructing equivariant supernatural vector bundles and equivariant pure Cohen–Macaulay modules using the Borel–Weil–Bott theorem. So in this post, I’d like to say something about what this theorem says and next time discuss how it can be used. I learned the stuff on Bott’s theorem from Jerzy Weyman’s book Cohomology of Vector Bundles and Syzygies [warning: there are some mistakes in the statement of Bott's theorem for general reductive groups]. Bott’s theorem is usually stated for a reductive group, but for concreteness we’ll stick with the general linear group, since that’s all we’ll need.

The setup is as follows: k denotes some field of characteristic 0, $G = \mathbf{GL}(n)$ , B is the subgroup of upper triangular matrices, and T is the subgroup of diagonal matrices. Then G/B is the complete flag variety whose k-points correspond to maximal flags $V_\bullet = (0 \subset V_1 \subset \cdots \subset V_n = k^n)$ where $\dim V_i = i$ . We are interested in realizing representations of G as cohomology groups of line bundles over G/B. But first we’ll state the relative version of the theorem.

For this, let X be any variety (really, X could be any scheme) and $\mathcal{E}$ be a vector bundle of rank n over X. We can construct the relative flag variety as follows. For any affine open set U inside of X where we get a trivialization $\mathcal{E}|_U = U \times k^n$ , let $\mathrm{Fl}(\mathcal{E},U)$ be the complete flag variety of $k^n$ . Using the gluing data of $\mathcal{E}$ , we can also glue together the spaces $\mathrm{Fl}(\mathcal{E}, U) \times U$ (on intersections, we have an isomorphism $k^n \to k^n$ , and this defines an isomorphism of complete flag varieties) to get a scheme $\mathrm{Fl}(\mathcal{E})$ which maps to X, call the map h. In fact, h is a locally trivial fibration whose fibers are complete flag varieties. To define the analogues of $R_i$ , consider the vector bundle $h^*(\mathcal{E}) = \{(x, F_\bullet, y) \mid x \in X,\ F_\bullet \in \mathrm{Fl}(\mathcal{E})|_x,\ y \in \mathcal{E}|_x\}$ on $\mathrm{Fl}(\mathcal{E})$ . We define $\mathcal{R}_i$ to be the subbundle of $h^*(E)$ consisting of points where $y \in F_i$ . Note that if we took X to be a single point (in particular Spec(k)), then $\mathcal{E}$ is just a k-vector space, and $\mathrm{Fl}(\mathcal{E})$ is the usual flag variety. So we can also define, in analogy, $\mathcal{L}(\alpha) = \mathcal{R}_1^{-\alpha_1} \otimes (\mathcal{R}_2 / \mathcal{R}_1)^{-\alpha_2} \otimes \cdots \otimes (\mathcal{E} / \mathcal{R}_n)^{-\alpha_n}$ .

Now let W be the Weyl group of G, i.e., the normalizer of T in G quotiented by T, in our case W is the symmetric group on n letters. In fact, W is a finite Coxeter group, and hence is equipped with a length function, denoted $\ell$ . For a permutation w, $\ell(w)$ is just the minimal number of simple transpositions $i \leftrightarrow i+1$ you need when expressing w as a product of them. So W has a natural action on the set of weights of T, and we define a dotted action of W on the weight lattice as follows. First, let $2\rho$ be the sum of the positive roots (i.e., the weights that appear when T acts on B via conjugation). In our case, we have $\rho = (n-1, n-2, \dots, 1, 0)$ . Then for $w \in W$ and $\alpha$ a weight, we set $w^\bullet(\alpha) = w(\alpha + \rho) - \rho$ . Let $K_\beta$ denote the Weyl functor, or co-Schur functor associated to the weight $\beta$ . I’ll describe its construction in a later post. For now, it’s a functorial way of constructing, from E, an irreducible representation of $\mathbf{GL}(E)$ of highest weight $\beta$ . We have the following theorem:

Theorem. With the notation above, one of two mutually exclusive cases occurs:

There exists a nonidentity element $w \in S_n$ such that $w^\bullet(\alpha) = \alpha$ . In this case, all higher direct images $\mathrm{R}^ih_*(\mathcal{L}(\alpha))$ vanish.
Otherwise, there is a unique $w \in S_n$ such that $\beta = w^\bullet(\alpha)$ is a dominant weight. In this case, if $i \ne \ell(w)$ , then $\mathrm{R}^ih_*^i(\mathcal{L}(\alpha)) = 0$ , and $\mathrm{R}^{\ell(w)}h_*(\mathcal{L}(\alpha)) = K_\beta(\mathcal{E}^*)$ is an equivariant bundle whose fibers are the dual of the irreducible representation of $\mathbf{GL}(n)$ with highest weight $\beta$ .

This gives us the cohomology for line bundles over G/B since when X is a point (and hence $\mathcal{E}$ is just a vector space), the higher direct images become cohomology groups.

If we assume this theorem, we can use the Leray spectral sequence to extract some information about vector bundles on partial flag varieties. Let $b = (0=b_0, b_1, \dots, b_r)$ be a sequence of increasing numbers, and let $\mathrm{Fl}(b; \mathcal{E})$ be the relative partial flag variety (i.e., the fibers are flags $F_\bullet = (F_{i_1} \subset F_{i_2} \subset \cdots \subset F_{i_r} \subset k^n)$ where $\dim F_{i_j} = b_j$ . So we still have the tautological bundles $\mathcal{R}_{b_i}$ . For each r>j>0, let $\alpha^j$ be a nonincreasing sequence of numbers of length $b_j - b_{j-1}$ , and write $\alpha = (\alpha^1, \dots, \alpha^r)$ (this is an n-tuple of numbers, not an r-tuple of sequences). Then we can define a bundle $\mathcal{V}(\alpha) = K_{\alpha^1}(\mathcal{R}_{b_1})^* \otimes \cdots \otimes K_{\alpha^r}(\mathcal{R}_{b_r} / \mathcal{R}_{b_{r-1}})^*$ . Now Bott’s theorem is the same as above except $\mathrm{Fl}(b; \mathcal{E})$ and $\mathcal{V}(\alpha)$ replace $\mathrm{Fl}(\mathcal{E})$ and $\mathcal{L}(\alpha)$ , respectively.

To deduce this version, note that we have a forgetful morphism $f \colon \mathrm{Fl}(\mathcal{E} \to \mathrm{Fl}(b; \mathcal{E})$ , which is a locally trivial fibration whose fibers are products of complete flag varieties. On $\mathrm{Fl}(\mathcal{E})$ , we can construct the line bundle $\mathcal{L}(\alpha)$ as before. Using the relative version of Bott’s theorem and induction on the number of terms that we forget, we see that $f_*(\mathcal{L}(\alpha)) = \mathcal{V}(\alpha)$ , and its higher direct images vanish because we assumed that each $\alpha^j$ was a dominant weight. Letting $h \colon \mathrm{Fl}(b; \mathcal{E}) \to X$ be the structure map, we can also use the relative Bott’s theorem to calculate $\mathrm{R}^i(hf)_*(\mathcal{L}(\alpha))$ . Alternatively, we have the Leray spectral sequence $\mathrm{E}_2^{p,q} = (\mathrm{R}^ph_* \circ \mathrm{R}^qf_*)(\mathcal{L}(\alpha)) \Rightarrow \mathrm{R}^{p+q}(hf)_*(\mathcal{L}(\alpha))$ . But we just showed that these terms are 0 for q=0, so the spectral sequence degenerates and we conclude that $\mathrm{R}^ph_*(\mathcal{V}(\alpha)) = \mathcal{R}^p(hf)_*(\mathcal{L}(\alpha))$ .

I’ll end this post with what this all says on projective space, and next time I’ll discuss how we can use this special case to construct equivariant pure modules and supernatural vector bundles.

Over projective space $\mathbf{P}(E) \cong \mathbf{P}^{n-1}_k$ , the fiber of the tautological bundle $\mathcal{R}$ over a point (which is a line in $E \cong k^n$ ) is the simply the vectors lying in that line. This is also the total space of the dual of the Serre twisting sheaf $\mathcal{O}_{\mathbf{P}(E)}(-1)$ . Given the inclusion $\mathcal{R} \subset E \times \mathbf{P}(E)$ , the quotient will be called $\mathcal{Q}$ and called the tautological quotient. Then our bundles of interest are of the form $K_u\mathcal{R} \otimes K_\alpha \mathcal{Q} = (K_\alpha \mathcal{Q})(-u)$ where u is an integer and $\alpha = (\alpha_1 \ge \alpha_2 \ge \cdots \ge \alpha_{n-1})$ . Note that this is the same $\mathcal{V}(\gamma)$ where $\gamma = (-u, -\alpha_{n-1}, ..., -\alpha_1)$ by duality. Hence the theorem says

Theorem. With the notation above, one of two mutually exclusive cases occurs:

There exists a nonidentity element $w \in S_n$ such that $w^\bullet(\alpha_1, \dots, \alpha_{n-1}, u) = \alpha$ . In this case, all cohomology groups $\mathrm{H}^i(\mathbf{P}(E); K_\alpha\mathcal{Q}(-u))$ vanish.
Otherwise, there is a unique $w \in S_n$ such that $\beta = w^\bullet(\alpha_1, \dots, \alpha_{n-1}, u)$ is nondecreasing. In this case, if $i \ne \ell(w)$ , then $\mathrm{H}^i(\mathbf{P}(E); (K_\alpha \mathcal{Q})(-u)) = 0$ , and $\mathrm{H}^{\ell(w)}(\mathbf{P}(E); (K_\alpha \mathcal{Q})(-u)) = K_\beta(E)$ is an irreducible representation of $\mathbf{GL}(E)$ with highest weight $\beta$ .