4.1 General case

The homogeneous space $G/N$ is a principal bundle $G/N \rightarrow G/MAN = \mathbb {F}$ with structural group $MA$ because N is normal in $MAN$ and $MAN/N \approx MA$ . The left action of G on $G/N$ commutes with the right action of $MA$ , meaning that G acts on the principal bundle $G/N \rightarrow \mathbb {F}$ by bundle automorphisms.

If $y_{0} = 1N$ is the origin of $G/N$ , then the orbit $Ky_{0}$ is the homogeneous space $K/\left ( K \cap N \right ) = K/\{1\} = K$ . This orbit is a subbundle (reduction) of $G/N$ with structural subgroup M, and $K \rightarrow K/M$ is the canonical projection associated with the action of K on $\mathbb {F}$ .

The right action of A allows us to write $G/N = \left ( Ky_{0} \right ) A$ , and in this way, $G/N$ becomes a principal A-bundle over the K-orbit $Ky_{0}$ . This principal bundle can be viewed in terms of homogeneous spaces through the canonical fibration $G/N \rightarrow G/AN = K$ , which is a principal bundle with structural group $AN/N = A$ .

This bundle is trivial (unlike the bundle over $\mathbb {F}$ ) because, by the Iwasawa decomposition, $G/N$ is diffeomorphic to $K \times A$ , which allows us to write

$$\begin{align*}G/N = K \times A, \end{align*}$$

that is, $G/N$ is identified with $K \times A$ , where in this identification, the origin $1 \cdot N$ of $G/N$ is identified with $\left ( 1,1 \right ) \in K \times A$ .

The bundle $K \rightarrow K/M = \mathbb {F}$ has a K-invariant (left) connection where the horizontal space $\mathcal {H}_{1}$ at the origin is the space orthogonal to the isotropy algebra $\mathfrak {m}$ :

$$\begin{align*}\mathcal{H}_{1} = \sum_{\alpha} \mathfrak{k}_{\alpha} \end{align*}$$

with $\mathfrak {k}_{\alpha } = \mathfrak {k} \cap \left ( \mathfrak {g}_{\alpha } \oplus \mathfrak {g}_{-\alpha } \right )$ .

This connection extends to a connection on $G/N \rightarrow \mathbb {F}$ since $K \rightarrow \mathbb {F}$ is a subbundle. The extension is also K-invariant. The connection form is denoted by $\omega _{K}$ . The same notation is used for both bundles $K \rightarrow \mathbb {F}$ and $G/N \rightarrow \mathbb {F}$ . On K, this form takes values in $\mathfrak {m}$ , while on $G/N$ , the form $\omega _{K}$ takes values in $\mathfrak {m} \oplus \mathfrak {a}$ .

The curvature $\Omega _{K}$ of the connection on K is given by $\Omega _{K}(A, B) = [A, B]_{\mathfrak {m}}$ , where A and B are left-invariant vector fields on K, and the subscript $\mathfrak {m}$ indicates the component in the direction of $\mathfrak {m}$ . Note that $\Omega _{K}$ is a 2-form on K with values in $\mathfrak {m}$ .

The curvature on $G/N$ is obtained from the curvature on K by right translations by elements of A. In general, a curvature satisfies the property

$$\begin{align*}R_{h}^{\ast}\Omega_{K} = \mathrm{Ad}\left( h^{-1} \right) \Omega_{K}. \end{align*}$$

The group A commutes with $\mathfrak {m}$ , so $\mathrm {Ad}\left ( h^{-1} \right )$ is the identity on $\mathfrak {m}$ . Hence, on $G/N$ , the curvature $\Omega _{K}$ also takes values in $\mathfrak {m}$ .

The component in the direction of $\mathfrak {a}$ of the curvature $\Omega _{K}$ vanishes. The same does not happen with the $\mathfrak {a}$ -component of the connection $\omega _{K}$ . Next, we will derive an expression for this component by composing the connection with linear functionals of $\mathfrak {a}$ .

If $\lambda \in (\mathfrak {m} \oplus \mathfrak {a})^{\ast }$ is a linear functional, then the composition $\lambda (\omega _{K})$ is a 1-form on $G/N$ . The following proposition provides a description in terms of the Cartan–Killing form for a composition $\lambda _{H}(\omega _{K})$ when $\lambda _{H}(\cdot ) = \langle H, \cdot \rangle $ is the linear functional dual to $H \in \mathfrak {m} \oplus \mathfrak {a}$ . The formula presented in the next proposition holds for an $MA$ -invariant functional $\lambda _{H}$ , which occurs if H is in the center of $\mathfrak {m} \oplus \mathfrak {a}$ (e.g., if $H \in \mathfrak {a}$ or any H in the case of a complex algebra, because in this case $\mathfrak {m} \oplus \mathfrak {a}$ is a Cartan subalgebra and thus abelian).

Proposition 4.1 Consider $G = KAN$ with its corresponding Iwasawa decomposition, and for $g \in G$ , write $g = k h n$ for the Iwasawa decomposition of g, where $k \in K$ , $h \in A$ , and $n \in N$ . Let $Z \in \mathfrak {g}$ , and denote by $\widetilde {Z}$ the vector field on $G / N$ induced by Z. Let $x_{0} = 1 \cdot N$ be the origin of $G / N$ . Then,

(4.2) $$ \begin{align} \lambda_{H}(\omega_{K}(\widetilde{Z}(gx_0))) = \langle \mathrm{Ad}(k) H, Z \rangle. \end{align} $$

Proof The formula is checked first at the origin $x_0 = 1 \cdot N$ and with $g = 1$ . Write the decomposition $Z = X + A + Y \in \mathcal {H}_1 \oplus (\mathfrak {m} \oplus \mathfrak {a}) \oplus \mathfrak {n}^{+}$ where, as mentioned above, $\mathcal {H}_1 = \sum _{\alpha } \mathfrak {k}_{\alpha }$ and $\mathfrak {k} = \mathcal {H}_1 \oplus \mathfrak {m}$ . By definition, $\omega _K(\widetilde {Z}(x_0)) = A$ , and thus $\lambda _H \omega _K(\widetilde {Z}(x_0)) = \lambda _H(A) = \langle H, A \rangle $ .

However, in the decomposition $\mathfrak {g} = \mathfrak {k} \oplus \mathfrak {a} \oplus \mathfrak {n}^{+}$ , the $\mathfrak {a}$ component is orthogonal (with respect to the Cartan–Killing form) to $\mathcal {H}_1 \oplus \mathfrak {n}^{+}$ . Therefore, $\langle H, A \rangle = \langle H, Z \rangle $ , showing that $\lambda _H \omega _K(\widetilde {Z}(x_0)) = \langle H, Z \rangle $ .

Still at the origin $x_0$ , but with $g = n \in N$ , the equality continues to hold because in the Iwasawa decomposition of g, the component $k = 1$ .

Now, consider a point $x = g x_0$ with $g = ma \in MA$ . This point is also given by $R_g(x_0)$ , since the left and right actions of $MA$ on the fiber over the origin coincide. Furthermore, the equality $\widetilde {Z}(R_g x_0) = (R_g)_{\ast }(\widetilde {Z}(x_0))$ holds because the left action of G commutes with the right action of $MA$ . Therefore,

(4.3) $$ \begin{align} \omega_K(\widetilde{Z}(gx_0)) = \omega_K((R_g)_{\ast}(\widetilde{Z}(x_0))) = (R_g^{\ast} \omega_K)(\widetilde{Z}(x_0)). \end{align} $$

But $R_g^{\ast } \omega _K = \mathrm {Ad}(g^{-1}) \omega _K$ , and since $\lambda _H$ is $MA$ -invariant, it follows that $\lambda _H(R_g^{\ast } \omega _K) = \lambda _H(\omega _K)$ for $g \in MA$ . Hence,

(4.4) $$ \begin{align} \lambda_H(\omega_K(\widetilde{Z}(gx_0))) = \lambda_H(\omega_K(\widetilde{Z}(x_0))) = \langle H, Z \rangle, \end{align} $$

where the last equality follows from the calculation at $x_0$ . If $g_1$ is such that $x = g_1 x_0 = g x_0$ , then $g_1 = g n = ma n$ with $n \in N^{+}$ , and as in the case of the origin, multiplication on the right by $n \in N^{+}$ does not affect the terms of the equality. This concludes the proof of (4.2) for points in the fiber over the origin $x_0$ .

For other fibers, the result is obtained by translating by $k \in K$ . From $\widetilde {Z}(kx) = k_{\ast } \widetilde {\mathrm {Ad}(k^{-1}) Z}(x)$ , it follows that

$$\begin{align*}\omega_K(\widetilde{Z}(kx)) = \omega_K(\widetilde{\mathrm{Ad}(k^{-1}) Z}(x)). \end{align*}$$

Therefore, if x is in the fiber over the origin, then

$$\begin{align*}\lambda_H(\omega_K(\widetilde{Z}(kx))) = \lambda_H(\omega_K(\widetilde{\mathrm{Ad}(k^{-1}) Z}(x))) = \langle H, \mathrm{Ad}(k^{-1}) Z \rangle = \langle \mathrm{Ad}(k) H, Z \rangle, \end{align*}$$

which proves the equality (4.2) for points in the fiber over $kx_0$ . Since K acts transitively on the fibers, this concludes the proof.

The second term $\langle \mathrm {Ad}(k)H, Z \rangle $ in (4.2) is the “height function” $f_Z$ with respect to Z, defined by the Cartan–Killing form on the adjoint orbit $\mathrm {Ad}(K)H$ . More precisely, for each $Z \in \mathfrak {g}$ , we define the height function on the adjoint orbit $f_Z : \mathrm {Ad}(K)H \to \mathbb {C}$ by

$$\begin{align*}f_Z(\mathrm{Ad}(k)H) = \langle \mathrm{Ad}(k)H, Z \rangle. \end{align*}$$

If $H \in \mathfrak {a}$ , then this adjoint orbit is a flag manifold of $\mathfrak {g}$ . However, if $H \in \mathfrak {m}$ , then $\mathrm {Ad}(K) H$ is an adjoint orbit of K in $\mathfrak {k}$ . For example, if $\mathfrak {g}$ is a complex algebra, then $\mathfrak {m} = i\mathfrak {h}_{\mathbb {R}}$ is the imaginary part of the Cartan subalgebra, and if $H \in i\mathfrak {h}_{\mathbb {R}}$ , then the K-adjoint orbit is also a flag manifold of $\mathfrak {g}$ .

From these observations about the K-adjoint orbits and formula (4.2), we obtain the following vanishing properties of $\lambda _H \omega _K$ .

Proof In both cases, Z is orthogonal (with respect to the Cartan–Killing form) to the adjoint orbit of H, and therefore the results follow from

$$ \begin{align*} \lambda_H \left( \omega_K \left( \widetilde{Z} (x) \right) \right) = \langle \mathrm{Ad}(k) H, Z \rangle \end{align*} $$

as stated in (4.2).

The hypothesis that H is in the center of $\mathfrak {m} \oplus \mathfrak {a}$ is automatically satisfied in cases where $\mathfrak {m}$ , and therefore $\mathfrak {m} \oplus \mathfrak {a}$ , is an abelian algebra. In these cases, the above corollary ensures that the $\mathfrak {m}$ or $\mathfrak {a}$ component of $\omega _K$ vanishes, and therefore the connection takes values in the other component.

4.2 Specializing to $\mathbb {F} \times \mathbb {F} = \mathbb {F} \times \mathbb {F}^{\ast }$ , the maximal f lag of $G \times G$

Definition 4.4 Consider the Iwasawa decomposition of $G \times G$ , given by $(K \times K)(A \times A)(N^{+} \times N^{-})$ . The maximal flag manifold is

(4.5) $$ \begin{align} \mathbb{F} \times \mathbb{F} = (G \times G) / (M \times M)(A \times A)(N^{+} \times N^{-}), \end{align} $$

which is based at the point $(x_{0}, x_{w_{0}}) \in \mathbb {F} \times \mathbb {F}$ , where $w_{0}$ denotes the principal involution of the Weyl group $\mathcal {W}$ .

The homogeneous space $(G \times G) / (N^{+} \times N^{-})$ is diffeomorphic to $G/N^{+} \times G/N^{-} = (K \times A) \times (K \times A)$ . Moreover, the projection

(4.6) $$ \begin{align} (G \times G) / (N^{+} \times N^{-}) \rightarrow \mathbb{F} \times \mathbb{F}^{\ast} = (G \times G) / (M \times M)(A \times A)(N^{+} \times N^{-}) \end{align} $$

defines a principal $(M \times M)(A \times A)$ -bundle.

Lemma 4.6 The projection

(4.7) $$ \begin{align} (G \times G) / (N^{+} \times N^{-}) \rightarrow (G \times G) / (A \times A)(N^{+} \times N^{-}) = K \times K \end{align} $$

is a principal bundle with fiber $A \times A$ , which is trivial since

(4.8) $$ \begin{align} (G \times G) / (N^{+} \times N^{-}) = (K \times K) \times (A \times A), \end{align} $$

where this identification is chosen so that the origin $(1,1)\cdot (N^{+} \times N^{-})$ corresponds to $((1,1),(1,1))$ .

Proposition 4.7 The diagonal action of G on $(G \times G)/(N^{+} \times N^{-})$ , given by

$$\begin{align*}g(g_{1}N^{+}, g_{2}N^{-}) = (gg_{1}N^{+}, gg_{2}N^{-}), \end{align*}$$

has trivial isotropy at the origin $o = (1,1)(N^{+} \times N^{-})$ . Hence, the orbit $G \cdot o$ is diffeomorphic to G.

For $z = (m_{1}h_{1}, m_{2}h_{2})(N^{+} \times N^{-})$ with $m_{i}h_{i} \in MA$ , the isotropy group of the G-action remains trivial, and its orbits project onto the open orbit $G/MA = G \cdot (x_{0}, x_{w_{0}}) \subset \mathbb {F} \times \mathbb {F}$ .

The restriction of the principal bundle

$$\begin{align*}(G \times G) / (N^{+} \times N^{-}) \rightarrow \mathbb{F} \times \mathbb{F} \end{align*}$$

to the open orbit $G \cdot (x_{0}, x_{w_{0}})$ is an $(MA) \times (MA)$ -principal bundle whose G-orbits (under the diagonal action) are diffeomorphic to G. Each orbit remains invariant under the right action of

(4.9) $$ \begin{align} \Delta_{MA} = \{(h,h):h \in MA\}, \end{align} $$

and hence these orbits form subbundles with structural group $\Delta _{MA}$ .

The group $G \times G$ acts on $K \times K$ , which is identified with $G \times G / \left ( AN^{+} \times AN^{-} \right )$ . Therefore, $G \approx \Delta _{G}$ acts on $K \times K$ via the diagonal action (restriction of the action of $G \times G$ to the diagonal $\Delta _{G}$ ). If $\overline {w}_{0}$ is a representative in $\mathrm {Nor}_{K} \left ( A \right )$ of the principal involution $w_{0}$ , then the orbit $G \cdot \left ( 1, \overline {w}_{0} \right )$ is open in $K \times K$ and projects to the unique open orbit in $\mathbb {F} \times \mathbb {F}^{\ast }$ .

The action of $G \times G$ on $\left ( K \times K \right ) \times \left ( A \times A \right ) = \left ( G \times G \right ) / \left ( N^{+} \times N^{-} \right )$ decomposes into the action on $K \times K$ and the cocycle $\rho ^{\#}: \left ( G \times G \right ) \times \left ( K \times K \right ) \rightarrow A \times A$ defined by

(4.10) $$ \begin{align} \rho^{\#} \left( \left( g, h \right), \left( k, u \right) \right) = \left( \rho \left( g, k \right), \rho \left( h, u \right) \right), \end{align} $$

where $\rho : G \times K \rightarrow A$ is, as before, the cocycle defined by

(4.11) $$ \begin{align} gu = x \rho \left( g, u \right) n \in KAN^{+}, \end{align} $$

which factors to a cocycle over $\mathbb {F}$ given by $\rho \left ( g, x \right ) = \rho \left ( g, u \right )$ .

With the cocycle notation (cf. Equation (4.11)), the action of $G \times G$ on $\left ( K \times K \right ) \times \left ( A \times A \right )$ is determined by

(4.12) $$ \begin{align} \left( g, h \right) \cdot \left( \left( k, u \right), \left( 1, 1 \right) \right) = \left( \left( g \cdot k, h \cdot u \right), \left( \rho \left( g, k \right), \rho \left( h, u \right) \right) \right), \end{align} $$

where in the first-ordered pair on the right-hand side, $\cdot $ denotes the action of G on K. (For an arbitrary point $\left ( \left ( k, u \right ), \left ( a, b \right ) \right )$ , the action is obtained by right invariance.)

In particular, the diagonal action of G is given by

(4.13) $$ \begin{align} \left( g, g \right) \cdot \left( \left( k, u \right), \left( 1, 1 \right) \right) = \left( \left( g \cdot k, g \cdot u \right), \left( \rho \left( g, k \right), \rho \left( g, u \right) \right) \right). \end{align} $$

If $\overline {w}_{0}$ is a representative in $\mathrm {Nor}_{K}(A)$ of the principal involution $w_{0}$ of the Weyl group, then the orbit $G \cdot \left ( \left ( 1, \overline {w}_{0} \right ), \left ( 1, 1 \right ) \right )$ under the diagonal action is diffeomorphic to G because the isotropy group is $N^{+} \cap N^{-} = \{1\}$ .

An orbit of this type is the subset of $\left ( K \times K \right ) \times \left ( A \times A \right )$ given by

$$ \begin{align*} G \cdot \left( \left( 1, \overline{w}_{0} \right), \left( 1, 1 \right) \right) &= \left\{ \left( \left( g \cdot 1, g \cdot \overline{w}_{0} \right), \left( \rho \left( g, 1 \right), \rho \left( g, \overline{w}_{0} \right) \right) \right) : g \in G \right\} \\ &= \left\{ \left( \left( g \cdot 1, g \cdot \overline{w}_{0} \right), \left( \rho \left( g, x_{0} \right), \rho \left( g, x_{w_{0}} \right) \right) \right) : g \in G \right\}. \end{align*} $$

That is, by writing the Iwasawa decompositions $g = kan^{+} \in KAN^{+}$ and $g = uhn^{-} \in KAN^{-}$ , one obtains $a = \rho \left ( g, 1 \right )$ , $h = \rho \left ( g, \overline {w}_{0} \right )$ , $k = g \cdot 1$ , and $u = g \cdot \overline {w}_{0}$ . Therefore, the orbit is given by the points

(4.14) $$ \begin{align} \left( \left( k, u \right), \left( \rho \left( g, x_{0} \right), \rho \left( g, x_{w_{0}} \right) \right) \right) \qquad g = kan^{+} = uhn^{-}, \end{align} $$

and these ordered pairs also provide the diffeomorphism between G and the orbit $G \cdot \left ( \left ( 1, \overline {w}_{0} \right ), \left ( 1, 1 \right ) \right )$ from the chosen origin.

Summarizing, we have proven the following.

Proposition 4.8 The orbit $G \cdot ((1,\overline {w}_0),(1,1))$ is

$$ \begin{align*} G \cdot ((1,\overline{w}_0),(1,1)) &= \left\{ \big( (g \cdot 1, g \cdot \overline{w}_0), (\rho(g,x_0), \rho(g,x_{w_0})) \big) : g \in G \right\}, \end{align*} $$

and is diffeomorphic to G.

Since an orbit $G \cdot \left ( \left ( 1, \overline {w}_{0} \right ), \left ( 1, 1 \right ) \right )$ is diffeomorphic to G, its admits the Cartan decomposition $G = SK$ . Therefore, the orbit is diffeomorphic to the product $S \times K$ .

To see this decomposition as subsets of $\left ( K \times K \right ) \times \left ( A \times A \right )$ contained in the orbit, first consider the K component, whose orbit is

$$ \begin{align*} K \cdot \left( \left( 1, \overline{w}_{0} \right), \left( 1, 1 \right) \right) = \left\{ \left( \left( k, k \overline{w}_{0} \right), \left( 1, 1 \right) \right) : k \in K \right\} = \left( K \times K \overline{w}_{0} \right) \times \left\{ \left( 1, 1 \right) \right\}. \end{align*} $$

The image of an arbitrary element $g = sk \in SK$ is obtained by applying k and then s to the origin $\left ( \left ( 1, \overline {w}_{0} \right ), \left ( 1, 1 \right ) \right )$ . Using the expression in k and the equality (4.13), the result is

$$ \begin{align*} \left( s, s \right) \cdot \left( \left( k, k \right), \left( 1, 1 \right) \right) = \left( \left( s \cdot k, s \cdot k \right), \left( \rho \left( s, k \right), \rho \left( s, k \overline{w}_{0} \right) \right) \right). \end{align*} $$

In particular, if $s = h = e^{H} \in A$ and $\left ( k, k \right ) = \left ( 1, 1 \right )$ , then

$$ \begin{align*} \left( h, h \right) \cdot \left( \left( 1, 1 \right), \left( 1, 1 \right) \right) &= \left( \left( 1, 1 \right), \left( \rho \left( h, 1 \right), \rho \left( h, \overline{w}_{0} \right) \right) \right) \\ &= \left( \left( 1, 1 \right), \left( h, \overline{w}_{0}^{-1} h \overline{w}_{0} \right) \right) \\ &= \left( \left( 1, 1 \right), \left( e^{H}, e^{w_{0} H} \right) \right). \end{align*} $$

Let $\theta $ be the Cartan involution on G given by the Cartan decomposition $G = KS$ .

An element $g \in G$ viewed in the open orbit of $\left ( K \times K \right ) \times \left ( A \times A \right )$ is given by

$$ \begin{align*} g = \left( \left( k, u \right), \left( \rho \left( g, x_{0} \right), \rho \left( g, x_{w} \right) \right) \right), \end{align*} $$

where $g = k \rho \left ( g, x_{0} \right ) n^{+} = u \rho \left ( g, x_{w} \right ) n^{-}$ are Iwasawa decompositions.

Applying $\theta $ to these Iwasawa decompositions, we obtain

$$ \begin{align*} \theta \left( g \right) = \theta \left( k \right) \theta \left( \rho \left( g, x_{0} \right) \right) \theta \left( n^{+} \right) = \theta \left( u \right) \theta \left( \rho \left( g, x_{w} \right) \right) \theta \left( n^{-} \right). \end{align*} $$

The Cartan involution $\theta $ is the identity on K and is the inverse on S, and moreover, $\theta \left ( N^{+} \right ) = N^{-}$ and $\theta \left ( N^{-} \right ) = N^{+}$ . Thus, we obtain the Iwasawa decompositions

$$ \begin{align*} \theta \left( g \right) &= k \rho \left( g, x_{0} \right)^{-1} \theta \left( n^{+} \right) \in KAN^{-}, \\ \theta \left( g \right) &= u \rho \left( g, x_{w} \right)^{-1} \theta \left( n^{-} \right) \in KAN^{+}. \end{align*} $$

Hence, $\theta \left ( g \right )$ viewed in the open orbit of $\left ( K \times K \right ) \times \left ( A \times A \right )$ is

$$ \begin{align*} \theta \left( g \right) = \left( \left( u, k \right), \left( \rho \left( g, x_{w} \right)^{-1}, \rho \left( g, x_{0} \right)^{-1} \right) \right). \end{align*} $$

This means that the involution $\theta $ viewed in the open orbit is given by

(4.15) $$ \begin{align} \theta : \left( \left( k, u \right), \left( h_{1}, h_{2} \right) \right) \longmapsto \left( \left( u, k \right), \left( h_{2}^{-1}, h_{1}^{-1} \right) \right). \end{align} $$

In summary, we have the following lemma.

Lemma 4.9 Let $\theta $ be the Cartan involution on G given by $G=KS$ . For $g=k\rho (g,x_0)n^{+}=u\rho (g,x_w)n^{-}$ , the involution acts as

(4.16) $$ \begin{align} \theta(g)=((u,k),(\rho(g,x_w)^{-1},\rho(g,x_0)^{-1})). \end{align} $$

Consequently, on the open orbit,

(4.17) $$ \begin{align} \theta : ((k,u),(h_1,h_2)) \mapsto ((u,k),(h_2^{-1},h_1^{-1})). \end{align} $$

An application of formula (4.2) provides the following expressions for compositions of linear functionals with the connection $\omega _{K \times K}$ on the bundle $G \rightarrow G / MA$ .

Proposition 4.10 Let $H_{1}, H_{2}$ be elements of the center of $\mathfrak {m} \oplus \mathfrak {a}$ and consider the linear functional $\lambda _{H_{1}, H_{2}}$ on $\left ( \mathfrak {m} \oplus \mathfrak {a} \right ) \times \left ( \mathfrak {m} \oplus \mathfrak {a} \right )$ defined by

$$ \begin{align*} \lambda_{H_{1}, H_{2}} \left( A, B \right) = \langle H_{1}, A \rangle + \langle H_{2}, B \rangle. \end{align*} $$

Then the composite $\lambda _{H_{1}, H_{2}} \left ( \omega _{K \times K} \right )$ is given by

$$ \begin{align*} \lambda_{H_{1}, H_{2}} \left( \omega_{K \times K} \left( \widetilde{Z}_{1} \left( g_{1} x_{0} \right), \widetilde{Z}_{2} \left( g_{2} x_{w} \right) \right) \right) = \langle \mathrm{Ad} \left( k_{1} \right) H_{1}, Z_{1} \rangle + \langle \mathrm{Ad} \left( k_{2} \right) H_{2}, Z_{2} \rangle \end{align*} $$

where $\left ( g_{1}, g_{2} \right ) \in G \times G$ , $\left ( Z_{1}, Z_{2} \right ) \in \mathfrak {g} \times \mathfrak {g}$ , and $g_{1} = k_{1} a_{1} n^{+} \in KAN^{+}$ , $g_{2} = k_{2} a_{2} n^{-} \in KAN^{-}$ are the components of $g_{1}$ and $g_{2}$ with respect to the opposite Iwasawa decompositions.

In particular, taking $Z_{1} = Z_{2}$ , the vector field $\left ( \widetilde {Z}, \widetilde {Z} \right )$ , $Z \in \mathfrak {g}$ , comes from the diagonal action of G. By identifying G with the orbit $G \cdot o$ , $\left ( \widetilde {Z}, \widetilde {Z} \right )$ becomes the right-invariant vector field $Z^{r}$ , and the 1-form $\omega _{K \times K}$ on G is given by $\omega _{K \times K} \left ( Z^{r} \right ) = \omega _{K \times K} \left ( \widetilde {Z}, \widetilde {Z} \right )$ .

Corollary 4.11 If $H_{1}, H_{2}$ belong to the center of $\mathfrak {m} \oplus \mathfrak {a}$ , then the composition $\lambda _{H_{1}, H_{2}} \left ( \omega _{K \times K} \right )$ evaluated at the right-invariant field $Z^{r}$ on G, which coincides with $\left ( \widetilde {Z}, \widetilde {Z} \right )$ , $Z \in \mathfrak {g}$ , is given by

$$ \begin{align*} \lambda_{H_{1}, H_{2}} \left( \omega_{K \times K} \left( Z^{r} \left( g x_{0}, g x_{w} \right) \right) \right) &= \lambda_{H_{1}, H_{2}} \left( \omega_{K \times K} \left( \widetilde{Z} \left( g x_{0} \right), \widetilde{Z} \left( g x_{w} \right) \right) \right) \\ &= \langle \mathrm{Ad} \left( k_{1} \right) H_{1}, Z \rangle + \langle \mathrm{Ad} \left( k_{2} \right) H_{2}, Z \rangle \end{align*} $$

where $g = k_{1} a_{1} n^{+} \in KAN^{+}$ and $g = k_{2} a_{2} n^{-} \in KAN^{-}$ are the components of g with respect to the opposite Iwasawa decompositions. Note that $G = KAN^{+} = KAN^{-}$ .

5.1 Generalities

Let $\pi : T^{\ast }M \rightarrow M$ be a cotangent bundle.

1. (1) The Liouville 1-form $\Lambda $ on $T^{\ast }M$ is defined by $\Lambda _{p}(v) = p(\pi _{\ast } v)$ , and the canonical symplectic form is $\Gamma = d\Lambda $ .

2. (2) If g is a diffeomorphism of M, its lift is given by $g^{\#} \cdot \alpha = \alpha \circ \left ( dg^{-1} \right )_{gx}$ for $\alpha \in T^{\ast }_x M$ . The lift $g^{\#}$ is also a diffeomorphism and is linear on each fiber. The forms $\Lambda $ and $\Gamma $ are invariant under $g^{\#}$ , that is, $\left ( g^{\#} \right )^{\ast } \Lambda = \Lambda $ and $\left ( g^{\#} \right )^{\ast } \Gamma = \Gamma $ . The invariance of $\Gamma $ under $g^{\#}$ means that $g^{\#}$ is a symplectic diffeomorphism. (A converse also holds: a diffeomorphism of $T^{\ast }M$ is a lift if and only if $\Lambda $ is invariant.)

3. (3) If X is a vector field on M, its lift $X^{\#}$ is the vector field whose flow is $\left (X_{t}\right )^{\#}$ , where $X_{t}$ is the flow of X.

   The vector field $X^{\#}$ is the Hamiltonian field (with respect to $\Gamma $ ) of the function $h(\alpha ) = \alpha (X(x))$ , where $\alpha \in T_x^{\ast }M$ .

   By definition, $h(\alpha ) = \Lambda (X^{\#})$ , that is,

   $$ \begin{align*} X^{\#} = \mathrm{ham}(\Lambda(X^{\#})). \end{align*} $$

   Since the lifts $\left (X_{t}\right )^{\#}$ leave $\Lambda $ invariant, the Lie derivative $L_{X^{\#}} \Lambda = 0$ . (The converse also holds: $L_{Y} \Lambda = 0$ if and only if Y is a lift, $Y = X^{\#}$ .)

4. (4) Given a function f on M, define $f^{\#} = f \circ \pi $ , which is constant along the fibers. The Hamiltonian vector field $\mathrm {ham}(f^{\#})$ is the “vertical” vector field that is constant on each fiber and is given by

   $$ \begin{align*} \mathrm{ham}(f^{\#})(p) = p + df_{x}, \qquad p \in T_x^{\ast}M. \end{align*} $$

   The trajectories of $\mathrm {ham}(f^{\#})$ are the straight lines $p + t \, df_{x}$ contained in the fibers.

   By general properties of Hamiltonian fields,

   $$ \begin{align*} \Gamma \left( X^{\#},\mathrm{ham}f^{\#}\right) =\left( \mathrm{ham} f^{\#}\right) \Lambda \left( X^{\#}\right) =-X^{\#}f^{\#}=-Xf. \end{align*} $$

5. (5) Lie group actions: Let G be a Lie group acting on M, where the infinitesimal action of the Lie algebra $ \mathfrak {g}$ is denoted by $X\in \mathfrak {g}\mapsto \widetilde {X}$ . The lifts $g^{\#}$ , $g\in G$ , define an action of G on $T^{\ast }M$ . This action is symplectic (with respect to $\Gamma $ ) because the lifts $g^{\#}$ are symplectic diffeomorphisms.

   The corresponding infinitesimal action is given by the lifts $X^{\#}$ of the vector fields $\widetilde {X}$ on M with $X\in \mathfrak {g}$ .

   A lift $X^{\#}$ is the Hamiltonian field (with respect to $\Gamma $ ) of the function $\Lambda \left ( X^{\#}\right )$ . This implies that the action of G on $T^{\ast }M$ is Hamiltonian (as well as symplectic).

   The moment map $\mu :T^{\ast }M\rightarrow \mathfrak {g}^{\ast }$ is defined by

   (5.1) $$ \begin{align} \mu \left( p\right) \left( X\right) =\Lambda _{p}\left( X^{\#}\left( p\right) \right) \qquad p\in T^{\ast }M,~X\in \mathfrak{g}. \end{align} $$

   An alternative description of the moment map $\mu $ can be given more directly from the action: denote the action on M by $\phi :G\times M\rightarrow M$ with $\phi \left ( g,x\right ) =g\cdot x$ . For each $x\in M$ , let $\phi _{x}:G\rightarrow M$ be defined by $ \phi _{x}\left ( g\right ) =g\cdot x$ . With this notation, $\widetilde {X} \left ( x\right ) =\left ( d\phi _{x}\right ) _{1}\left ( X\right ) $ , $X\in \mathfrak {g}=T_{1}G$ . If $p\in T_{x}^{\ast }M$ , then

   $$ \begin{align*} \Lambda _{p}\left( X^{\#}\left( p\right) \right) =p\left( \widetilde{X} \left( x\right) \right) =p\circ \left( d\phi _{x}\right) _{1}\left( X\right). \end{align*} $$

   However, by the definition of the adjoint linear transformation, $ p\circ \left ( d\phi _{x}\right ) _{1}=\left ( d\phi _{x}\right ) _{1}^{\ast }\left ( p\right ) $ .

   Therefore, in terms of the action $\phi $ , the moment map is given by

   (5.2) $$ \begin{align} \mu _{\left\vert T_{x}^{\ast }M\right. }=\left( d\phi _{x}\right) _{1}^{\ast}. \end{align} $$

6. (6) Equivariance of the moment map: The moment map on $T^{\ast }M$ defined via the Liouville form $\Lambda _{p}\left ( X^{\#}\left ( p\right ) \right ) $ is equivariant, meaning

   $$ \begin{align*} \mu \left( g^{\#}p\right) =\mathrm{Ad}^{\ast }\left( g\right) \mu \left( p\right) \qquad p\in T^{\ast }M,~g\in G. \end{align*} $$

   Indeed, if $p\in T_{x}^{\ast }M$ , then $g^{\#}p\in T_{gx}^{\ast }M$ , and we have

   $$ \begin{align*} \mu \left( g^{\#}p\right) \left( X\right) &=\Lambda _{g^{\#}p}\left( X^{\#}\left( g^{\#}p\right) \right) =g^{\#}p\left( \widetilde{X}\left( gx\right) \right) =p\left( g_{\ast }^{-1}\widetilde{X}\left( gx\right) \right) =p( \widetilde{\mathrm{Ad}\left( g^{-1}\right)} \\ & X\left( x\right) ) =\mu \left( p\right) \left( \mathrm{Ad}\left( g^{-1}\right) X\right) =\left( \mathrm{Ad}^{\ast }\left( g\right) \mu \left( p\right) \right) \left( X\right). \end{align*} $$

   The equivariance of $\mu $ implies that the image of an orbit $ G^{\#}\cdot p$ under the action of G on $T^{\ast }M$ is the coadjoint orbit $\mathrm {Ad}^{\ast }\left ( G\right ) \cdot \mu \left ( p\right ) \subset \mathfrak {g}^{\ast }$ . Furthermore, if $\omega _{\mu \left ( p\right ) }$ is the KKS symplectic form on the orbit $\mathrm {Ad} ^{\ast }\left ( G\right ) \cdot \mu \left ( p\right ) $ , then the restriction of the symplectic form $\Gamma $ to $G^{\#}\cdot p$ is the pullback $\mu ^{\ast }\omega _{\mu \left ( p\right ) }$ .

5.2 Flag manifolds

The maximal flag manifold $\mathbb {F}$ is given by a K-adjoint orbit $ \mathrm {Ad}\left ( K\right ) H_{0}=K/M$ . The tangent bundle $T\mathbb {F}$ and cotangent bundle $T^{\ast }\mathbb {F}$ are realized as the G-adjoint orbit $\mathrm {Ad}\left ( G\right ) H_{0}=G/MA$ with same $H_{0}$ . For $\mathrm {Ad}\left ( K\right ) H_{0}$ to indeed represent the maximal flag manifold, $H_{0}$ must be chosen appropriately, which occurs, for instance, in the following cases.

1. (1) $H_{0}\in \mathfrak {a}^{+}$ is a regular real element.

2. (2) $\mathfrak {g}$ is a complex Lie algebra where $\mathfrak {m}\oplus \mathfrak {a}=\mathfrak {h}$ is a (complex) Cartan subalgebra with $ \mathfrak {a}=\mathfrak {h}_{\mathbb {R}}$ as the real part and $\mathfrak {m}=i \mathfrak {h}_{\mathbb {R}}$ as the imaginary part. In this case, an orbit $\mathrm {Ad}\left ( K\right ) H_{0}$ represents the maximal flag manifold if $H_{0}$ is regular (real or not).

In both $\mathrm {Ad}\left ( K\right ) H_{0}=K/M$ and $\mathrm {Ad}\left ( G\right ) H_{0}=G/MA$ , the origin $x_{0}$ is precisely $H_{0}$ .

The fibers of the bundles are in bijection with affine subspaces.

1. (1) The fiber of $T\mathbb {F}$ over $\mathrm {Ad}\left ( k\right ) H_{0}$ is $ \mathrm {Ad}\left ( k\right ) \left ( H_{0}+\mathfrak {n}^{-}\right ) $ , $k\in K$ . The bijection associates $\mathrm {Ad}\left ( k\right ) \left ( H_{0}+Y\right )$ , $Y\in \mathfrak {n}^{-}$ , with the tangent vector $\widetilde {Y}\left ( kx_{0}\right )$ .

2. (2) The fiber of $T^{\ast }\mathbb {F}$ over $\mathrm {Ad}\left ( k\right ) H_{0}$ is $\mathrm {Ad}\left ( k\right ) \left ( H_{0}+\mathfrak {n}^{+}\right )$ , $k\in K$ . The bijection associates $\mathrm {Ad}\left ( k\right ) \left ( H_{0}+X\right )$ , $X\in \mathfrak {n}^{+}$ , with the linear functional $p\in T_{kx_{0}}^{\ast }$ defined by $p\left ( \widetilde {Y}\left ( kx_{0}\right ) \right ) =\langle X,Y\rangle $ , where $Y\in \mathfrak {n}^{-}$ and $\langle \cdot ,\cdot \rangle $ is the Cartan–Killing form of $\mathfrak {g}$ .

The affine subspaces of $\mathfrak {g}$ contained in $\mathrm {Ad}\left ( G\right ) H_{0}$ , which appear as fibers of $T\mathbb {F}$ and $T^{\ast } \mathbb {F}$ , are determined by the following Lie algebra decompositions.

First, consider the decomposition $\mathfrak {g}=\mathfrak {m} \oplus \mathfrak {a}\oplus \mathfrak {n}^{+}\oplus \mathfrak {n}^{-}$ arising from the root spaces. More generally, for $k\in K$ , consider the conjugated decomposition:

$$ \begin{align*} \mathfrak{g}=\mathfrak{m}^{k}\oplus \mathfrak{a}^{k}\oplus \left( \mathfrak{n }^{+}\right) ^{k}\oplus \left( \mathfrak{n}^{-}\right) ^{k}, \end{align*} $$

where the superscript indicates conjugation by k ( $\mathfrak {m}^{k}=\mathrm {Ad}\left ( k\right ) \mathfrak {m}$ , etc.).

With this notation, the affine subspaces are given by:

$$ \begin{align*} \mathrm{Ad}\left( k\right) \left( H_{0}+\mathfrak{n}^{-}\right) =\mathrm{Ad} \left( k\right) H_{0}+\left( \mathfrak{n}^{-}\right) ^{k}, \!\!\!\qquad \mathrm{Ad}\left( k\right) \left( H_{0}+\mathfrak{n}^{+}\right) =\mathrm{Ad}\left( k\right) H_{0}+\left( \mathfrak{n}^{+}\right) ^{k}, \end{align*} $$

where the first is identified with the tangent space $T_{x}\mathbb {F}$ and the second with $T_{x}^{\ast }\mathbb {F}$ when $x=\mathrm {Ad}\left ( k\right ) H_{0}$ .

The identifications of the tangent and cotangent spaces with affine subspaces of $\mathfrak {g}$ are used below to describe the Liouville and canonical forms of the cotangent bundle $T^{\ast }\mathbb {F}$ . These forms are obtained from the moment map $\mu $ associated with the Hamiltonian action of G on $T^{\ast }\mathbb {F}$ , derived from the lift of the action $\phi : G \times \mathbb {F} \rightarrow \mathbb {F}$ on $\mathbb {F}$ . As shown in (5.2), the following equality holds:

$$ \begin{align*} \mu _{\vert T_{x}^{\ast }M }=\left( d\phi _{x}\right)_{1}^{\ast}. \end{align*} $$

The isotropy algebra of the action $\phi $ at the point $x = kx_{0}$ is given by $\mathfrak {m}^{k} \oplus \mathfrak {a}^{k} \oplus \left ( \mathfrak {n}^{+} \right )^{k}$ . Consequently, if $Z \in \mathfrak {g}$ decomposes as

$$ \begin{align*} Z = Z_{\mathfrak{m}} + Z_{\mathfrak{a}} + Z_{\mathfrak{n}^{+}} + Z_{\mathfrak{n}^{-}} \in \mathfrak{m}^{k} \oplus \mathfrak{a}^{k} \oplus \left( \mathfrak{n}^{+} \right)^{k} \oplus \left( \mathfrak{n}^{-} \right)^{k}, \end{align*} $$

then $\widetilde {Z}(x) = \widetilde {Z}_{\mathfrak {n}^{-}}(x)$ , with $x = kx_{0}$ . Thus,

$$ \begin{align*} \left( d\phi_{x} \right)_{1}(Z) = \widetilde{Z}_{\mathfrak{n}^{-}}(x) \qquad Z \in \mathfrak{g}. \end{align*} $$

Identifying $T_{x}\mathbb {F}$ , where $x = kx_{0}$ , with $\left ( \mathfrak {n}^{-} \right )^{k} = \mathrm {Ad}(k)\mathfrak {n}^{-}$ , the linear map $\left ( d\phi _{x} \right )_{1}$ corresponds to the projection $\mathrm {proj}_{\left ( \mathfrak {n}^{-} \right )^{k}}$ of $\mathfrak {g}$ onto $\left ( \mathfrak {n}^{-} \right )^{k}$ , with respect to the decomposition

$$ \begin{align*} \mathfrak{g} = \mathfrak{m}^{k} \oplus \mathfrak{a}^{k} \oplus \left( \mathfrak{n}^{+} \right)^{k} \oplus \left( \mathfrak{n}^{-} \right)^{k}. \end{align*} $$

The moment map $\mu $ is the adjoint $\left ( d\phi _{x} \right )_{1}^{\ast }$ , that is,

$$\begin{align*}\mu\left( \alpha \right)\left( Z \right) = \alpha \circ \left( d\phi_{x} \right)_{1}^{\ast} \left( \widetilde{Z}\left( x \right) \right) = \alpha \circ \left( d\phi_{x} \right)_{1}^{\ast} \left( \widetilde{Z_{\left( \mathfrak{n}^{-} \right)^{k}}} \left( x \right) \right), \quad \alpha \in T_{x}^{\ast}\mathbb{F}, \, Z \in \mathfrak{g}. \end{align*}$$

Therefore, $\mu \left ( \alpha \right )\left ( Z \right ) = \mu \left ( \alpha \right )\left ( Z_{\left ( \mathfrak {n}^{-} \right )^{k}} \right )$ .

Identifying $\mathfrak {g}^{\ast }$ with $\mathfrak {g}$ using the Cartan–Killing form $\langle \cdot , \cdot \rangle $ , the subspace $\mathfrak {m}^{k} \oplus \mathfrak {a}^{k}$ vanishes on $\left ( \mathfrak {n}^{-} \right )^{k}$ , so $\mu \left ( \alpha \right )$ lies in the nilpotent algebra $\left ( \mathfrak {n}^{+} \right )^{k}$ .

It is now possible to interpret the moment map using the identification of $T^{\ast }\mathbb {F}$ with $G/MA=\mathrm {Ad}\left (G\right ) H_{0}$ . This adjoint orbit is the union

$$ \begin{align*} \mathrm{Ad}\left( G\right) H_{0} = \bigcup_{k\in K} \mathrm{Ad}\left(k\right) \left( H_{0} + \mathfrak{n}^{+} \right). \end{align*} $$

An element $H_{0}^{k} + X^{k} = \mathrm {Ad}\left (k\right ) \left (H_{0} + X\right )$ , with $X \in \mathfrak {n}^{+}$ , is viewed as a linear functional on the tangent space $T_{x}\mathbb {F} = \mathrm {Ad}\left ( k \right ) \left (H_{0} + \mathfrak {n}^{-}\right )$ , where $x = \mathrm {Ad}\left (k\right ) x_{0}$ , and is expressed as $\langle X, Y \rangle $ with $Y \in \mathfrak {n}^{-}$ .

Summarizing, we have the following proposition.

Proposition 5.1 Let $\mu :T^{\ast }\mathbb {F} \rightarrow \mathfrak {g}^{\ast }$ be the moment map induced by the linear action of G on $T^{\ast }\mathbb {F}$ . Using the identifications $\mathfrak {g}^{\ast } \approx \mathfrak {g}$ (via the Cartan–Killing form $\langle \cdot , \cdot \rangle $ ) and $T^{\ast }\mathbb {F} \approx \mathrm {Ad}\left (G\right ) H_{0}$ , the map $\mu $ is given by

(5.3) $$ \begin{align} \mu \left( H_{0}^{k} + X^{k} \right)\left(Z\right) = \langle X^{k}, Z \rangle, \end{align} $$

where the superscript k indicates $X^{k} = \mathrm {Ad}\left (k\right ) X$ , and similarly for other terms.

This expression for the moment map $\mu $ provides descriptions of the canonical symplectic form on $T^{\ast }\mathbb {F}$ .

One description is given in terms of nilpotent orbits in $\mathfrak {g}$ . A key property of an equivariant moment map (as is the case here) is that the image of a group orbit is a coadjoint orbit of the representation. Furthermore, the restriction of the symplectic form to the orbit is the pullback of the KKS form on the coadjoint orbit.

In this case, the orbits of the linear action of G on $T^{\ast }\mathbb {F}$ are given by

$$ \begin{align*} G^{\#} \cdot p = \{ p \circ dg_{x}^{-1} : g \in G \} \qquad p \in T_{x}^{\ast }\mathbb{F}, \end{align*} $$

with $x \in \mathbb {F}$ being arbitrary, as the action of G on $\mathbb {F}$ is transitive.

By identifying $T^{\ast }\mathbb {F} \approx \mathrm {Ad}\left (G\right ) H_{0}$ , an element $\alpha \in T_{x_{0}}^{\ast }\mathbb {F}$ , where $x_{0} = H_{0}$ , is given by $\alpha = H_{0} + X$ , with $X \in \mathfrak {n}^{+}$ .

From (5.3), we have $\mu \left (H_{0} + X\right ) = \langle X, \cdot \rangle $ . Hence, the following description of the canonical symplectic form $\Gamma $ on $T^{\ast }\mathbb {F}$ is obtained.

Proposition 5.2 Consider the identification $T^{\ast }\mathbb {F} \approx \mathrm {Ad}\left (G\right ) H_{0}$ . Then, the canonical symplectic form $\Gamma $ on $T^{\ast }\mathbb {F}$ , restricted to the orbit $G^{\#} \cdot \left (H_{0} + X\right )$ , $X \in \mathfrak {n}^{+}$ , is the pullback of the KKS form on the coadjoint orbit of $\langle X, \cdot \rangle $ .

Via the identification $\mathfrak {g}^{\ast } \approx \mathfrak {g}$ , the coadjoint orbit of $\langle X, \cdot \rangle $ corresponds to the adjoint orbit of the nilpotent element X.

Explicitly, if $p = \mathrm {Ad}\left (k\right )\left (H_{0} + X\right )$ , then

(5.4) $$ \begin{align} \Gamma_{p}\left(\widetilde{Y}, \widetilde{Z}\right) = \langle X, \left[Y, Z\right] \rangle \qquad Y, Z \in \mathfrak{g}, \end{align} $$

where $\widetilde {Y} = \mathrm {ad}\left (Y\right )$ and $\widetilde {Z} = \mathrm {ad}\left (Z\right )$ are the linear vector fields on $\mathfrak {g}$ defined by Y and Z.

Another application of the moment map formula (5.3) is that it provides an expression for the Liouville form $\Lambda $ in terms of the difference $\omega - \omega _{K \times K}$ , where $\omega $ and $\omega _{K \times K}$ are the connections that are G-invariant and $K \times K$ -invariant, respectively. From this, an expression for the canonical symplectic form in terms of the difference $\Omega - \Omega _{K \times K}$ of the respective curvatures is obtained.

For this interpretation, it must be assumed that $H_{0}$ lies in the center of $\mathfrak {m} \oplus \mathfrak {a}$ so that $\lambda _{H_{0}}$ is $MA$ -invariant. The condition $\mathrm {Ad}\left (K\right ) H_{0} = \mathbb {F}$ is still used, as $\mathrm {Ad}\left (G\right ) H_{0}$ must be identified with $T^{\ast }\mathbb {F}$ .

The relationship between $\Lambda $ and $\mu $ is given by $\mu \left (Z\right ) = \Lambda \left (Z^{\#}\right )$ , where $Z \in \mathfrak {g}$ . Combining this equality with (5.3) yields

(5.5) $$ \begin{align} \Lambda_{p}\left(Z^{\#}\left(p\right)\right) = \langle X^{k}, Z \rangle \qquad Z \in \mathfrak{g}, \end{align} $$

where $p = H_{0}^{k} + X^{k} = \mathrm {Ad}\left (k\right )\left (H_{0} + X\right )$ is an element of the cotangent space $T_{x}^{\ast }\mathbb {F} = \mathrm {Ad}\left (k\right )\left (H_{0} + \mathfrak {n}^{+}\right )$ , with $x = H_{0}^{k} = \mathrm {Ad}\left (k\right ) H_{0}$ .

The second term in the equality above can be rewritten as

(5.6) $$ \begin{align} \langle X^{k}, Z \rangle = \langle H_{0}^{k} + X^{k}, Z \rangle - \langle H_{0}^{k}, Z \rangle. \end{align} $$

This expression is interpreted in terms of connections as follows.

1. (1) Let $\omega $ be the G-invariant connection on $G \rightarrow G/MA$ . Then, the first term on the right-hand side of (5.6) is given by

   (5.7) $$ \begin{align} \langle H_{0}^{k} + X^{k}, Z \rangle = \lambda_{H_{0}}\left(\omega_{g} Z^{r}(g)\right), \end{align} $$
   where $Z^{r}$ is the right-invariant vector field defined by $Z \in \mathfrak {g}$ , $g \in G$ is any element such that $\mathrm {Ad}(g) H_{0} = H_{0}^{k} + X^{k}$ , and $\lambda _{H_{0}}(\cdot ) = \langle H_{0}, \cdot \rangle $ .

   Indeed, at the origin $H_{0}$ with $g = 1$ , the equality holds because $\omega _{1} Z^{r}(1)$ is the component of Z in the direction of $\mathfrak {m} \oplus \mathfrak {a}$ with respect to the decomposition $\mathfrak {g} = \mathfrak {m} \oplus \mathfrak {a} \oplus \mathfrak {n}^{+} \oplus \mathfrak {n}^{-}$ , which is orthogonal with respect to the Cartan–Killing form. At the origin, for $g \in MA$ , the equality still holds since $H_{0}^{k} + X^{k} = \mathrm {Ad}(g) H_{0} = H_{0}$ .

   For a general point, note that $\omega _{g}(Z^{r}(g)) = \omega _{1}((\mathrm {Ad}(g^{-1}) Z)^{r}(1))$ because $\omega $ is left-invariant. Thus,

   $$ \begin{align*} \lambda_{H_{0}}(\omega_{g} Z^{r}(g)) = \lambda_{H_{0}}\left(\omega_{1}((\mathrm{Ad}(g^{-1}) Z)^{r}(1))\right), \end{align*} $$
   which, using the equality at $H_{0}$ , becomes
   $$ \begin{align*} \lambda_{H_{0}}\left(\omega_{1}((\mathrm{Ad}(g^{-1}) Z)^{r}(1))\right) &= \langle H_{0}, \mathrm{Ad}(g^{-1}) Z \rangle = \langle \mathrm{Ad}(g) H_{0}, Z \rangle \\ &= \langle H_{0}^{k} + X^{k}, Z \rangle, \end{align*} $$
   if $\mathrm {Ad}(g) H_{0} = H_{0}^{k} + X^{k}$ . This concludes the proof of (5.7), which provides the first term on the right-hand side of (5.6) in terms of the G-invariant connection.

2. (2) The component $\langle H_{0}^{k}, Z \rangle = \langle \mathrm {Ad}(k) H_{0}, Z \rangle $ was computed in terms of the connection $\omega _{K \times K}$ in Corollary 4.11. The element $H_{0} \in \mathfrak {a}$ lies in the center of $\mathfrak {m} \oplus \mathfrak {a}$ , and thus $\left (H_{0}, 0\right )$ is in the center of $\left (\mathfrak {m} \oplus \mathfrak {a}\right ) \times \left (\mathfrak {m} \oplus \mathfrak {a}\right )$ . Consequently, Corollary 4.11 applies to the linear functional $\lambda _{H_{0}, 0}$ , showing that

   $$ \begin{align*} \lambda_{H_{0}, 0}\left(\omega_{K \times K}\left(Z^{r}(gx_{0}, gx_{w})\right)\right) &= \lambda_{H_{0}, 0}\left(\omega_{K \times K}\left(\widetilde{Z}(gx_{0}), \widetilde{Z}(gx_{w})\right)\right)\\ &= \langle \mathrm{Ad}(k) H_{0}, Z \rangle, \end{align*} $$
   where $g = khn^{+}$ are the components of g in the Iwasawa decomposition $G = KAN^{+}$ . This computes $\langle \mathrm {Ad}(k) H_{0}, Z \rangle $ in terms of the connection $\omega _{K \times K}$ .

The combination of the previous two items expresses the Liouville form of $ T^{\ast } \mathbb {F} $ in terms of the difference $ \omega - \omega _{K \times K} $ between the connections. If $ \lambda $ is an $ MA $ -invariant linear functional on $ \left ( \mathfrak {m} \oplus \mathfrak {a} \right )^2 $ , then $ \lambda \omega - \lambda \omega _{K \times K} $ is a right-invariant 1-form on the principal bundle $ \left ( G \times G \right ) / \left ( N^+ \times N^- \right ) \to G / MA = T^{\ast } \mathbb {F} $ , which vanishes on the vertical spaces (since this is the behavior of the difference between two connections). Therefore, $ \lambda \omega - \lambda \omega _{K \times K} $ is the pullback of a 1-form on $ G / MA $ . This 1-form on $ G / MA $ is denoted by $ \lambda ^{\ast } \left ( \omega - \omega _{K \times K} \right ) $ .

Proposition 5.3 Let the identification of $ T^{\ast } \mathbb {F} $ with $ G/MA $ be made by equating both to an adjoint orbit $ \mathrm {Ad}(G) H_0 $ , where $ H_0 $ is in the center of $ \mathfrak {m} \oplus \mathfrak {a} $ and regular in the sense that $ \mathrm {Ad}(K) H_0 = \mathbb {F} $ . Then, the Liouville form on $ T^{\ast } \mathbb {F} = G/MA $ is given by

$$ \begin{align*} \lambda_{H_0,0}^{\ast} \left( \omega - \omega_{K \times K} \right) \end{align*} $$

where, as before, $ \lambda _{H_0} (\cdot ) = \langle H_0, \cdot \rangle $ . Therefore, the canonical symplectic form on $ T^{\ast } \mathbb {F} $ is

$$ \begin{align*} \Gamma = \lambda_{H_0,0}^{\ast} \left( \Omega - \Omega_{K \times K} \right) \end{align*} $$

where $ \Omega $ and $ \Omega _{K \times K} $ are the curvatures. In this expression, the component $ \lambda _{H_0,0}^{\ast } \Omega = \Omega _{\lambda _{H_0}} $ is the KKS form of the adjoint orbit by $ H_0 $ .

As previously seen, the $ \mathfrak {a} \oplus \mathfrak {a} $ -component of the curvature $ \Omega _{K \times K} $ vanishes, that is, $ \Omega _{K \times K} $ takes values in $ \mathfrak {m} \oplus \mathfrak {m} $ . Therefore, if $ H_1 $ and $ H_2 $ are in $ \mathfrak {a} $ , then $ \lambda _{H_1, H_2}^{\ast } \Omega _{K \times K} = 0 $ . Taking, as in the previous proposition, $ (H_1, H_2) = (H_0, 0) $ where $ H_0 \in \mathfrak {a} $ , it follows that the canonical symplectic form reduces to $ \Gamma = \lambda _{H_0, 0}^{\ast } \Omega $ . The functional $ \lambda _{H_0, 0} $ restricted to the diagonal $ \{ (H, H) \in \mathfrak {a} \oplus \mathfrak {a} \} $ coincides with the functional $ \lambda _{H_0} $ on $ \mathfrak {a} $ . Therefore, $ \lambda _{H_0, 0}^{\ast } \Omega $ is seen as $ \lambda _{H_0}^{\ast } \Omega $ , which is in turn the KKS form of the adjoint orbit $ \mathrm {Ad}(G) H_0 $ . This shows that this KKS form is exact.

Corollary 5.4 If $ H_0 \in \mathfrak {a} $ is a regular real element, then the KKS form $ \kappa _{H_0} $ of the adjoint orbit $ \mathrm {Ad}(G) H_0 $ coincides with the canonical symplectic form $ \Gamma $ of $ T^{\ast } \mathbb {F} $ and is therefore an exact form. In terms of connections,

$$ \begin{align*} \kappa_{H_0} = d \left( \lambda_{H_0}^{\ast} \left( \omega - \omega_{K \times K} \right) \right) \end{align*} $$

where $ \omega $ is the $ G $ -invariant connection on $ G \to G / MA $ .

To conclude this section, we record some remarks regarding the preceding constructions.

1. (1) The fact that the canonical symplectic form $ \Gamma $ is written as the difference of two curvatures is in agreement with the Chern–Weil homomorphism, which shows that for an invariant functional $ \lambda $ and two curvatures $ \Omega _1 $ and $ \Omega _2 $ , the $ 2 $ -forms $ \lambda ^{\ast } \Omega _1 $ and $ \lambda ^\ast \Omega _2 $ are cohomologous. Therefore, the difference between them is an exact form, as is the case with the canonical form in a cotangent bundle.

2. (2) The cotangent bundle $ T^{\ast } \mathbb {F} $ has been identified with the adjoint orbit $ \mathrm {Ad}(G) H_0 $ (with $ \mathbb {F} = \mathrm {Ad}(K) H_0 $ ), where the fibers are the orbits of $ k N^+ k^{-1} $ with $ k \in K $ . The symmetric (dual) construction can be made with $ N^{-} $ instead of $ N^+ $ . For this, the origin of the flag must be shifted, now viewed as the orbit $ \mathrm {Ad}(K) (w_0 H_0) $ , where $ w_0 $ is the principal involution. This means that $ \mathbb {F} $ is now seen as $ \mathbb {F}^{\ast } $ . In this new origin, the Iwasawa decomposition $ G = KAN^{-} $ is considered, and the constructions work in the same way. This provides another diffeomorphism between $ T^{\ast } \mathbb {F} $ and the adjoint orbit (in fact, $ T^{\ast } \mathbb {F}^{\ast } $ ).

   In terms of the immersion of $ T^{\ast } \mathbb {F} $ in $ \mathbb {F} \times \mathbb {F}^{\ast } $ (as an open orbit), in the first diffeomorphism, the fibers are the fibers $ p_1^{-1}\{x\} $ with respect to the projection onto the first coordinate. In the second diffeomorphism, the fibers are $ p_2^{-1}\{y\} $ with respect to the projection onto the second coordinate.

   Changing the choice of diffeomorphism also changes the canonical symplectic form in the open orbit. For the first diffeomorphism, the linear functional $ \lambda _{H_0,0} $ was used in the proposition above. For the second choice, one simply takes the functional $ \lambda _{0, w_0 H_0} $ , when the canonical symplectic form is given by $ \lambda _{0, w_0 H_0}^{\ast } (\Omega - \Omega _{K \times K}) $ , with the Liouville form $ \lambda _{0, w_0 H_0}^{\ast } (\omega - \omega _{K \times K}) $ .

3. (3) The diffeomorphism between $ T^{\ast } \mathbb {F} $ and the adjoint orbit $ \mathrm {Ad}(G) H_0 $ with $ \mathbb {F} = \mathrm {Ad}(K) H_0 $ is such that the cotangent space $ T_{x}^{\ast } \mathbb {F} $ becomes the orbit $ N^k \cdot x $ , where $ x = \mathrm {Ad}(k) H_0 \in \mathbb {F} $ . In the diffeomorphism between $ T^{\ast } \mathbb {F} $ and the open orbit of the diagonal action $ G \cdot o = G \cdot (x_0, x_w) $ , the base $ \mathbb {F} $ becomes the $ K $ -orbit $ K \cdot (x_0, x_w) = \{(kx_0, kx_w) : k \in K\} $ , and a fiber $ T_{x}^{\ast } \mathbb {F} $ , $ x = (kx_0, kx_w) $ , is also the $ N^k $ -orbit $ N^k \cdot x $ . This orbit coincides with the fiber $ p_1^{-1} \{kx_0\} $ , where $ p_1 : G \cdot o \to \mathbb {F} $ is the projection onto the first coordinate.

   The fibers $ T_{x}^{\ast } \mathbb {F} $ are Lagrangian submanifolds with respect to $ \Gamma $ , as happens in any cotangent bundle. This can also be seen in the expression for $ \Gamma $ in terms of the difference $ \omega - \omega _{K \times K} $ of connections. To see this, take, for example, the fiber $ N^+ \cdot o $ over the origin, which is the projection of $ \pi (N^+) $ with respect to $ \pi : G \to G/MA $ . Given $ X, Y \in \mathfrak {n}^+ $ , the curvature $ \Omega $ computed on the left-invariant vector fields is $ \Omega (X^l, Y^l) $ , which is the $ \mathfrak {m} \oplus \mathfrak {a} $ component of the bracket $ [X, Y] $ . Hence, the curvature vanishes since $ [X, Y] \in \mathfrak {n}^- $ . Therefore, the component $ \lambda _{H_0,0}^{\ast } \Omega $ of the symplectic form vanishes on the fiber. The other component also vanishes on the fiber because $ \Omega _{K \times K} $ , computed on tangent vectors to the second coordinate, takes values in the second coordinate of $ (\mathfrak {m} \oplus \mathfrak {a})^2 $ , where the functional $ \lambda _{H_0,0} $ vanishes. The same argument works for any $ x = k(x_0, x_w) $ , by replacing $ N^+ $ with $ k N^+ k^{-1} $ . This shows that the fibers $ T_{x}^{\ast } \mathbb {F} = (\{x\} \times \mathbb {F}^{\ast }) \cap G \cdot o $ are Lagrangian submanifolds of the symplectic form $ \lambda _{H_0,0}^{\ast }(\Omega - \Omega _{K \times K}) $ .

   By switching the first and second coordinates, it is proven that the submanifolds $ T_{x}^{\ast } \mathbb {F} = (\mathbb {F} \times \{y\}) \cap G \cdot o $ are Lagrangian for $ \lambda _{0, w_0 H_0}^{\ast }(\omega - \omega _{K \times K}) $ .