Vector Bundles on Rational Homogeneous Spaces

In mathematics, Lie group–Lie algebra correspondence allows one to study Lie groups, which are geometric objects, in terms of Lie algebras, which are linear objects. Let $G$ be a semi-simple Lie group. Assume $V$ to be a nonzero finite dimensional complex vector space, fix a maximal torus $H\subset G$, and let $\varphi:G\rightarrow GL(V)$ be a representation of $G$. It is well known that $V$ decomposes into a direct sum of simultaneous eigenspaces

$$V=\bigoplus V_{\lambda},$$

where the direct sum run over $\lambda$ in the character group of $H$, which is the set of all holomorphic homomorphisms $\lambda$ from $H$ to $\mathbb{C}^{*}$, and

$$V_{\lambda}=\{v\in V|\varphi(h)=\lambda(h)v,~{}\text{for all}~{}h\in H\}.$$

Since $V$ is a finite dimensional vector space, we have $V_{\lambda}=0$ for all but finitely many values of $\lambda$. Those values of $\lambda$ for which $V_{\lambda}\neq 0$ are called the weights of $V$, and $V_{\lambda}$ is called the weight space.

In the Lie algebra side, let $\mathfrak{g}$ be the associated semi-simple Lie algebra of $G$. The maximal torus corresponds to $\mathfrak{h}\subset\mathfrak{g}$ which is an abelian subalgebra of maximal dimension, i.e. Cartan subalgebra. A holomorphic representation $\varphi:G\rightarrow GL(V)$ of a complex Lie group $G$ gives rise to a complex linear representation $\varphi_{\mathfrak{g}}:\mathfrak{g}\rightarrow\mathfrak{gl}(V)$ of the Lie algebra $\mathfrak{g}$ of $G$. Similarly, every finite dimensional representation of $\mathfrak{g}$ admits a decomposition

$$V=\bigoplus_{\lambda\in\{\lambda\in\mathfrak{h}^{\vee}|V_{\lambda}\neq 0\}}V_{\lambda},$$

where

$$V_{\lambda}=\{v\in V|[h,v]=\lambda(h)v,~{}\text{for all}~{}h\in\mathfrak{h}\}.$$

Those $\lambda$ are still called the weights of $V$, and $V_{\lambda}$ is called the weight space. If we apply the above decomposition to $V=\mathfrak{g}$ and $\varphi_{\mathfrak{g}}$ the adjoint representation, we have Cartan decomposition of $\mathfrak{g}$:

$$\mathfrak{g}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\mathfrak{h}^{\vee}\backslash\{0\}}\mathfrak{g}_{\alpha},$$

where

$$\mathfrak{g}_{\alpha}:=\{g\in\mathfrak{g}|[h,g]=\alpha(h)g,~{}\text{for all}~{}h\in\mathfrak{h}\}.$$

The elements $\alpha\in\mathfrak{h}^{\vee}\backslash\{0\}$ for which $\mathfrak{g}_{\alpha}\neq 0$ are called roots of $\mathfrak{g}$, and the set of these elements will be denoted by $\Phi$ and be called root system. For every $\alpha\in\Phi$, $\mathfrak{g}_{\alpha}$ is called root space which is one dimensional.

Fix a linear functional

$$f:\text{span}_{\mathbb{R}}\Phi\rightarrow\mathbb{R}$$

whose kernel does not intersect $\Phi$. Let

$$\Phi^{+}:=\{\alpha\in\Phi|f(\alpha)>0\}~{}\text{and}~{}\Phi^{-}:=\{\alpha\in\Phi|f(\alpha)<0\}.$$

Then $\Phi^{+}$ is called positive system of roots and $\Phi^{-}$ is called negative system of roots. Given such a positive system $\Phi^{+}$, we define the fundamental system $\Pi\subset\Phi^{+}$ as follows: $\alpha\in\Pi$ if and only if $\alpha\in\Phi^{+}$ and $\alpha$ cannot be expressed as the sum of two elements of $\Phi^{+}$. A non-zero representation $V$ of $\mathfrak{g}$ is called a highest weight representation if it is generated by a vector $v\in V_{\lambda}$ such that $gv=0$ for all $g\in\bigoplus_{\alpha\in\Phi^{+}}\mathfrak{g}_{\alpha}$. In this case, $v$ is called the highest weight vector, and $\lambda$ is the highest weight of $V$.

For each $\alpha\in\Phi$ there is a unique element $h_{\alpha}\in\mathfrak{h}$ such that

$$\alpha(h)=<h_{\alpha},h>~{}\text{for all}~{}h\in\mathfrak{h}.$$

The vectors $h_{\alpha}$ for $\alpha\in\Phi$ span $\mathfrak{h}$. We denote by $\mathfrak{h}_{\mathbb{R}}$ the set of all elements of form $\sum_{i=1}^{l}a_{i}h_{\alpha}$ for $a_{i}\in\mathbb{R}$.

The Killing form $<\alpha,\beta>:=tr(\text{ad}_{\alpha}\circ\text{ad}_{\beta})$ defines a nondegenerated bilinear form on $\mathfrak{h}$, where $\alpha,\beta\in\mathfrak{g}$ and ad is the adjoint representation. It can be shown that $\mathfrak{h}_{\mathbb{R}}^{\vee}$ is a Euclidean space with respect to $<,>$. Set $n:=dim_{\mathbb{C}}(\mathfrak{h})$ and $D:=\{1,2,\ldots,n\}$. We identify $D$ with the set of fundamental roots with respect to a choice of maximal torus $T$ and fixed Borel subgroup $B$. It is known that every fundamental system $\Pi\subset\Phi$ can form a basis of $\mathfrak{h}_{\mathbb{R}}^{\vee}$. Let $\Pi=\{\alpha_{1},\ldots,\alpha_{n}\}$ be a fundamental system. Then we define $A_{ij}$ by

$$A_{ij}=2\frac{<\alpha_{i},\alpha_{j}>}{<\alpha_{i},\alpha_{i}>}\in\mathbb{Z},\quad i,j=1,\ldots n.$$

The matrix $A=(A_{ij})$ is called the Cartan matrix of $\mathfrak{g}$.

The Dynkin diagram of $G$, which we denoted by $\mathcal{D}:=\mathcal{D}(G)$, is determined by the Cartan matrix. It consists of a graph whose set of nodes is $D$ and where the nodes $i$ and $j$ are joined by $A_{ij}A_{ji}$ edges. When two nodes $i$ and $j$ are joined by a double or triple edge, we add to it an arrow pointing to $i$ if $|A_{ij}|>|A_{ji}|$. We call $\alpha_{i}$ a short root of $\mathcal{D}$ and $\alpha_{j}$ a non-short (or long) root of $\mathcal{D}$. (Sometimes, for the sake of narrative convenience, we freely interchange the terminology "node" and "root".) One may prove that there is a one to one correspondence between isomorphism classes of semisimple Lie algebras and Dynkin diagrams of reduced root systems. Moreover, every reduced root system is a disjoint union of mutually orthogonal irreducible root subsystems, each of them corresponding to one of the connected finite Dynkin diagrams $A_{n}$, $B_{n}$, $C_{n}$, $D_{n}$ $(n\in\mathbb{Z}_{>0})$, $E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, $G_{2}$:

The connected components of the Dynkin diagram $\mathcal{D}$ determine the simple Lie groups that are factors of the semisimple Lie group $G$, each of them corresponding to one of the Dynkin diagrams above.

A closed subgroup $P$ of $G$ is called parabolic if the quotient space $G/P$ is complete, hence projective. A maximal connected solvable subgroup $B$ of $G$ is called a Borel subgroup. We fix a Cartan subalgebra $\mathfrak{h}$. Let

$$\mathfrak{b}=\mathfrak{h}\oplus\bigoplus_{\alpha\in\Phi^{-}}\mathfrak{g}_{\alpha}$$

be a fixed Borel subalgebra. It is easy to determine the parabolic subalgebras containing $\mathfrak{b}$. They are all of the form

$$\mathfrak{p}=\mathfrak{b}\oplus\bigoplus_{\alpha\in\Phi^{+}_{P}}\mathfrak{g}_{\alpha},$$

where $\Phi^{+}_{P}$ is a subset of $\Phi^{+}$ that is closed under the addition of roots. Hence the parabolic subalgebras containing $\mathfrak{b}$ lie in bijection with the subsets of $D$. For some subset $I$ of $D$, we write $\mathfrak{p}_{I}$ the parabolic subalgebra corresponding to $I$. We define $P_{I}$ by the parabolic subgroup of $G$ such that its Lie algebra is $\mathfrak{p}_{I}$. Therefore the parabolic subgroup $P_{I}$ corresponds to the subset $I$ (so a maximal parabolic subgroup is defined by a single root). Then $G/P_{I}$ has a minimal homogeneous embedding in projective space of the highest weigh module $V_{\lambda}$ of $G$ corresponding to the highest weight $\lambda=\sum_{i\in I}\omega_{i}$, where $\omega_{i}$ is the $i$-th fundamental weight dual to the roots $\alpha_{i}\in\Pi$, by a very ample line bundle $L_{\lambda}$.

It is well known that $G/P$ carries a transitive $G$-action, it is a smooth projective variety. Borel and Remmert’s classical theorem (⁴) states that a projective complex manifold which admits a transitive action of its automorphism group is a direct product of an abelian variety and a rational homogeneous space $G/P$, where $G$ is a semi-simple algebraic group and $P$ is a parabolic subgroup.

Every rational homogeneous space $G/P$ can be decomposed into a product

$$G/P\simeq G_{1}/P_{I_{1}}\times G_{2}/P_{I_{2}}\times\cdots\times G_{m}/P_{I_{m}}$$

of rational homogeneous spaces with simple algebraic group $G_{1},\cdots,G_{m}$. Each rational homogeneous space $G_{i}/P_{I_{i}}$, called the generalized flag manifold, only depends on the Lie algebra $\mathfrak{g}_{i}$ of $G_{i}$, which is classically determined by the marked Dynkin diagram (¹⁶). In the most common notation, we set $F_{I}:=G/P_{I}$ by marking on the Dynkin diagram $\mathcal{D}$ of $G$ the nodes corresponding to $I$. For instance, numbering the nodes of $A_{n}$, the usual flag manifold $F(d_{1},\ldots,d_{s};n+1)$ corresponds to the marking of $I=\{d_{1},\ldots,d_{s}\}$ (sometimes we omit the braces and just write as $P_{d_{1},\ldots,d_{s}}$).

The two extremal cases correspond to the generalized complete flag manifolds (all nodes marked), and the generalized Grassmannian (only one node marked).

In ¹⁶, the authors explicitly describe the lines through a point of a rational homogeneous space $G/P_{I}$, where $G$ is a simple Lie group. Let $j\in I$ and $N(j)$ be the set of nodes in $\mathcal{D}$ that are connected to $j$.

There is a similar statement for $C_{x}\subseteq\mathbb{P}T_{x}X$, the set of tangent directions to lines on $X$ passing through a fixed point $x$. It is a disjoint union of spaces of lines of class $\check{\alpha}$ through $x$.

Not only $\mathbb{P}^{1}$ but also all linear spaces can be read from the marked Dynkin diagrams.

Along this section we will work on uniform vector bundles on rational homogeneous spaces of Picard number one, i.e. generalized Grassmannians. Let $G$ be a simple Lie group and $D=\{1,2,\ldots,n\}$ be the set of nodes of the Dynkin diagram $\mathcal{D}$ of $G$. Denote by $P_{k}$ the parabolic subgroup of $G$ corresponds to the node $k$. Consider the generalized Grassmannian $\mathcal{G}=G/P_{k}$ or, for brevity, $\mathcal{D}/P_{k}$. Denote by $\mathcal{M}:=G/P_{N(k)}$ the generalized flag manifold defined by the marked Dynkin diagram $(\mathcal{D},N(k))$ and by $\mathcal{U}:=G/P_{k,N(k)}=G/(P_{N(k)}\cap P_{k})$ the universal family, which has a natural $\mathbb{P}^{1}$-bundle structure over $\mathcal{M}$, i.e. we have the natural diagram

(3.5)

Remarkably, $\mathcal{M}$ defined above is indeed an unsplit family of rational curves on $\mathcal{G}$. Given $x\in\mathcal{G}$, $\mathcal{M}_{x}=q(p^{-1}(x))$, which we call the special family of lines of class $\check{\alpha_{k}}$ through $x$, coincides with $H/P_{N(k)}$ by Remark 2.7, where the set of nodes of the Dynkin diagram $H$ is $D(H)=D\backslash{k}$.

When $k$ is an extremal node, that is, the subdiagram $\mathcal{D}(H)$ is connected. Remarkably, in the case $\mathcal{D}=D_{n}$, i.e. $G=SO(2n)$, since $G/P_{n-1}\cong G/P_{n}$, we only need to think about the extremal node $n$. Similarly, since $E_{6}/P_{1}\cong E_{6}/P_{6}$, we just consider the the extremal node $1$ in $E_{6}$. According to Theorem 2.6, $\mathcal{M}_{x}$ has the following possibilities:

• •

  Projective spaces or smooth quadrics,

• •

  Grassmannians,

• •

  Spinor varities,

• •

  $E_{6}/P_{6}$, $E_{7}/P_{7}$, $C_{3}/P_{3}$.

The possibilities are list in Table 1 below.