Gradient maps associated with actions of real reductive groups

Abstract

This thesis is concerned with the study of the action of a real reductive group $G$ on a real submanifold $X$ of a Kahler manifold $(Z, \omega)$ which is the restriction of a holomorphic action of a complex reductive Lie group $U^\mathbb{C}.$ Let $U$ be a compact connected Lie group with Lie algebra $\mathfrak{u}$ acting on $Z$ and preserving $\omega$. We assume that the $U$-action extends holomorphically to an action of the complexified group $U^\mathbb{C}$ and the $U$-action on $Z$ is Hamiltonian. Then there exists a $U$-equivariant momentum map $\mu : Z\to \mathfrak{u}$. If $G\subset U^\mathbb{C}$ is a closed subgroup such that the Cartan decomposition $U^\mathbb{C} = U\text{exp}(i\mathfrak{u})$ induces a Cartan decomposition $G = K\text{exp}(\mathfrak{p}),$ where $K = U\cap G$, $\mathfrak{p} = \mathfrak{g}\cap i\mathfrak{u}$ and $\mathfrak{g}=\mathfrak{k} \oplus \mathfrak{p}$ is the Lie algebra of $G$, there is a corresponding gradient map $\mu_\mathfrak{p} : X\to \mathfrak{p}$. Given an $\mathrm{Ad}(K)$-invariant scalar product on $\mathfrak{p}$, we obtain a Morse like function $f=\frac{1}{2}\parallel \mu_{\mathfrak p} \parallel^2$ on $X$. We study some properties of the gradient map $\mu_\mathfrak{p}$ and its norm square function $f$ and analyse the $G$-action on $X$ using the gradient map.