Harish-Chandra's Schwartz space

Definition

Let $G$ be a connected semisimple Lie group with finite centre, with Lie algebra $\mathfrak g$, and let $K$ be a maximal compact subgroup. Let $\mathfrak g = \mathfrak k \oplus \mathfrak p$ be the corresponding Cartan decomposition, and let $\norm{\cdot}$ be the norm on $\mathfrak p$ given by the Killing form. By the Cartan decomposition each $g \in G$ can be written $g = k \exp X$ with $k \in K$ and $X \in \mathfrak p$ unique; put $\sigma(g) = \norm{X}$. Let $\Xi = \varphi_0$ be the spherical function of parameter $0$ given by Harish-Chandra's formula. Elements $D$ of the universal enveloping algebra $U(\mathfrak g_\C)$ act on smooth functions on $G$ as left-invariant differential operators $f \mapsto f D$ and as right-invariant differential operators $f \mapsto D f$. Harish-Chandra's Schwartz space $\mathcal{C}(G)$ is the set of smooth functions $f \colon G \to \C$ such that $$\sup_{g \in G} \, \abs{(D f E)(g)} \, \Xi(g)^{-1} \, (1 + \sigma(g))^{r} < \infty$$ for all $D, E \in U(\mathfrak g_\C)$ and all $r \ge 0$.