I am reading notes by David Vogan on Unitary representations and Complex analysis (pdf / dvi).
The setting is as follows (page 23): Let $X$ be a $(\mathfrak{g},K)$-module and let $X(\mu)$ denote its $\mu$-isotypical component for $\mu\in\hat{K}$. On each $X(\mu)$ there is $K$-invariant scalar product $\|\ \|_\mu$.
Various globalizations of $X$ are given as spaces of sequences $(x(\mu))_{\mu\in\hat{K}}$ with certain conditions on $\| x(\mu)\| _\mu$. These conditions also lead to a canonical topology on the globalizations.
Vogan also mentions that even $X = \bigoplus_{\mu\in\hat{K}} X(\mu)$ itself and $\Pi_{\mu\in\hat{K}} X(\mu) = X^{-K}$ have canonical topologies.
Q1: How are these topologies defined?
Q2: Are the inclusions $X\subseteq X^{\text{glob}} \subseteq X^{-K}$ continuous with dense images?
Q3: Consider Hermitian symmetric case and complexifications. There is a parabolic subgroup of $G^\mathbb{C}$ given by $P=K^\mathbb{C}U$ with $U$ abelian such that the compact symmetric space $G/K$ is isomorphic to $G^\mathbb{C}/P$. Can one extend continuously the action of $K$ on to $P$ for $X$, $X^\text{glob}$ and $X^{-K}$?
