Lie groups and homogeneous spaces

In the Fifties Bott applied Morse theory with great success to the topology of Lie groups and homogeneous spaces. In [8] he showed how the diagram of a compact semisimple connected and simply connected group $G$ determines the integral homology of both the loop space $\Omega G$ and the flag manifold $G/T$ , where $T$ is a maximal torus.

Indeed, Morse theory gives a beautiful CW cell structure on $G/T$ , up to homotopy equivalence. To explain this, recall that the adjoint action of the group $G$ on its Lie algebra $\mathfrak{g}$ restricts to an action of the maximal torus $T$ on $\mathfrak{g}$ . As a representation of the torus $T$ , the Lie algebra $\mathfrak{g}$ decomposes into a direct sum of irreducible representations

$$\mathfrak{g}= \mathfrak{t}\oplus \sum E_{\alpha} ,$$

where $\mathfrak{t}$ is the Lie algebra of $T$ and each $E_{\alpha }$ is a $2$ -dimensional space on which $T$ acts as a rotation $e^{2\pi i \alpha (x)}$ , corresponding to the root $\alpha (x)$ on $\mathfrak{t}$ . The diagram of $G$ is the family of parallel hyperplanes in $\mathfrak{t}$ where some root is integral. A hyperplane that is the zero set of a root is called a root plane.

For example, for the group $G = \operatorname{SU}(3)$ and maximal torus

$$T=\left\{ \left. \begin{bmatrix} e^{2\pi i x_1} & & \\ & e^{2\p... ...} \end{bmatrix}\ \right\vert \ x_1+x_2+x_3 = 0, x_i \in \mathbb{R}\right\} .$$

the roots are $\pm(x_1 - x_2), \pm(x_1 - x_3) , \pm(x_2 - x_3)$ , and the diagram is the collection of lines in the plane $x_1+x_2+x_3=0$ in $\mathbb{R}^3$ as in Figure 3. In this figure, the root planes are the thickened lines.

Figure 3: The diagram of SU(3)

For $G=\operatorname{SU}(2)$ and

$$T= \left\{ \left. \begin{bmatrix} e^{2\pi i x} & 0 \cr 0 & e^{-2\pi i x} \end{bmatrix}\ \right\vert \ x \in\mathbb{R}\right\},$$

the Lie algebra $\mathfrak{t}$ is $\mathbb{R}$ , the roots are $\pm 2x$ , and the adjoint representation of of $G$ on $\mathfrak{g}= \mathbb{R}^3$ corresponds to rotations. The root plane is the origin.

Figure 4: The diagram of SU(2)

A point $B$ in $\mathfrak{t}$ is regular if its normalizer has minimal possible dimension, or equivalently, if its normalizer is $T$ . It is well known that a point $B$ in $\mathfrak{t}$ is regular if and only if it does not lie on any of the hyperplanes of the diagram. If $B$ is regular, then the stabilizer of $B$ under the adjoint action of $G$ is $T$ and so the orbit through $B$ is $G/T$ .

Choose another regular point $A$ in $\mathfrak{t}$ and define the function $f$ on $\operatorname{Orbit}(B) = G/T$ to be the distance from $A$ ; here the distance is measured with respect to the Killing form on $\mathfrak{g}$ . Let $\{ B_i\}$ be all the points in $\mathfrak{t}$ obtained from $B$ by reflecting about the root planes. Then Bott's theorem asserts that $f$ is a Morse function on $G/T$ whose critical points are precisely all the $B_i$ 's. Moreover, the index of a critical point $B_i$ is twice the number of times that the line segment from $A$ to $B_i$ intersects the root planes. This cell decomposition of Morse theory fits in with the more group-theoretic Bruhat decomposition.

For $G = \operatorname{SU}(3)$ and $T$ the set of diagonal matrices in $\operatorname{SU}(3)$ , the orbit $G/T$ is the complex flag manifold $\operatorname {F\ell }(1,2,3)$ , consisting of all flags

$$V_1 \subset V_2 \subset \mathbb{C}^3, \quad \dim_{\mathbb{C}} V_i = i.$$

Bott's recipe gives 6 critical points of index $0,2,2,4,4,6$ respectively on $G/T$ (See Figure 5). By the lacunary principle, the Morse function $f$ is perfect. Hence, the flag manifold $\operatorname {F\ell }(1,2,3)$ has the homotopy type of a CW complex with one 0 -cell, two $2$ -cells, two $4$ -cells, and one $6$ -cell. Its Poincaré polynomial is therefore

$$P_t(\operatorname{F\ell}(1,2,3)) = 1+2t^2 +2t^4 +t^6.$$

Figure 5: The flag manifold $\operatorname {F\ell }(1,2,3)$