Harish-Chandra's Schwartz space

In mathematical abstract harmonic analysis, Harish-Chandra's Schwartz space is a space of functions on a semisimple Lie group whose derivatives are rapidly decreasing, studied by Harish-Chandra. It is an analogue of the Schwartz space on a real vector space, and is used to define the space of tempered distributions on a semisimple Lie group.

Prerequisites

Length functions

Let $G$ be a topological group. A length on $G$ is a continuous function $l:G\mapsto [0,+\mathrm{\infty }[$, such that for all $g,g^{\prime }\in G$, $l(g{g}^{\prime })\le l(g)+l(g^{\prime })$. These functions are used to define spaces with rapidly decreasing functions on locally compact topological groups, since they are used to define polynomial weights to ensure rapid decay.

The $\sigma$ function

Let $G$ be a semisimple connected Lie group with Lie algebra $𝔤$, and let $K$ be its maximal compact subgroup. Then, the Cartan decomposition of $G$ allows one to state that the mapping $(k,X)\mapsto k\exp (X)$ ,$k\in K$ and $X\in 𝔭$ is an analytic diffeomorphism of $K\times 𝔭$ onto $G$.
Let $b$ be the Killing form of $𝔤$. Its restriction on $𝔭$ provides an euclidean norm $\Vert X\Vert$, $K$-invariant, and such that $⟨Z,(adZ{)}^{2}Y⟩=⟨(adZ{)}^{2}X,Y⟩$ for all $X,Y,Z\in 𝔭$, where $⟨\cdot ,\cdot ⟩$ is the inner product corresponding to $\Vert \cdot \Vert$.
The $\sigma$ function of $G$ is then defined as $\sigma (x)=\Vert X\Vert$ for $x=k\exp (X)$. It is proved that $\sigma$ is a length function. Of course, $\sigma$ is equal to $0$ on $K$, and for all $x\in G$, $\sigma (x^{-1})=\sigma (x)$.

Definition

In order to define the Harish-Chandra Schwartz space on a semisimple Lie group $G$, one uses the Harish-Chandra's Ξ function and the $\sigma$ function of $G$, defined in the former section. The Schwartz space on $G$ consists roughly of the functions all of whose derivatives are rapidly decreasing compared to Ξ. More precisely, let's suppose $G$ is connected, and let's define ${\nu }_{r,g_1,g_2}(f)=\underset{g\in G}{sup}|g_1\ast f(g)\ast g_2|{\mathrm{\Xi }}^{-1}(g)(1+\sigma (g){)}^r$ for all $r\in ℝ$, where $g_1$ and $g_2$ belong to the universal enveloping algebra $U𝔤$ of $𝔤$, $\ast$ denotes its action on $G$ as differential operators, and $r\in ℝ$.
The Harish-Chandra Schwartz space is defined as the subspace $S(G)$ of ${𝒞}^{\mathrm{\infty }}(G)$ such that $f\in S(G)$ if and only if ${\nu }_{r,g_1,g_2}(f)<+\mathrm{\infty }$ for every $g_1,g_2\in U𝔤$, $r\in ℝ$. Topologized with the set of such seminorms, it is a Fréchet space. Since $G$ is connected, the set of smooth functions with compact support ${𝒞}_c^{\mathrm{\infty }}(G)$ on $G$ is dense in $S(G)$.