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. ^[1] 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 ${\displaystyle G}$ be a topological group. A length on ${\displaystyle G}$ is a continuous function ${\displaystyle l:G\mapsto [0,+\infty [}$, such that for all ${\displaystyle g,g'\in G}$, ${\displaystyle l(gg')\leq l(g)+l(g')}$. ^[2] 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. ^{[1] [2]}

The ${\displaystyle \sigma }$ function

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

Definition

In order to define the Harish-Chandra Schwartz space on a semisimple Lie group ${\displaystyle G}$, one uses the Harish-Chandra's Ξ function and the ${\displaystyle \sigma }$ function of ${\displaystyle G}$, defined in the former section. The Schwartz space on ${\displaystyle G}$ consists roughly of the functions all of whose derivatives are rapidly decreasing compared to Ξ. More precisely, let's suppose ${\displaystyle G}$ is connected, and let's define ${\displaystyle \nu _{r,g_{1},g_{2}}(f)=\displaystyle \sup _{g\in G}{|g_{1}\ast f(g)\ast g_{2}|\Xi ^{-1}(g)(1+\sigma (g))^{r}}}$ for all ${\displaystyle r\in \mathbb {R} }$, where ${\displaystyle g_{1}}$ and ${\displaystyle g_{2}}$ belong to the universal enveloping algebra ${\displaystyle U{\mathfrak {g}}}$ of ${\displaystyle {\mathfrak {g}}}$, ${\displaystyle \ast }$ denotes its action on ${\displaystyle G}$ as differential operators, ^[3] and ${\displaystyle r\in \mathbb {R} }$.
The Harish-Chandra Schwartz space is defined as the subspace ${\displaystyle S(G)}$ of ${\displaystyle {\mathcal {C}}^{\infty }(G)}$ such that ${\displaystyle f\in S(G)}$ if and only if ${\displaystyle \nu _{r,g_{1},g_{2}}(f)<+\infty }$ for every ${\displaystyle g_{1},g_{2}\in U{\mathfrak {g}}}$, ${\displaystyle r\in \mathbb {R} }$. Topologized with the set of such seminorms, it is a Fréchet space. Since ${\displaystyle G}$ is connected, the set of smooth functions with compact support ${\displaystyle {\mathcal {C}}_{c}^{\infty }(G)}$ on ${\displaystyle G}$ is dense in ${\displaystyle S(G)}$. ^[1]

References

1. Harish-Chandra (1966), "Discrete series for semisimple Lie groups. II. Explicit determination of the characters", Acta Mathematica , 116: 1–111, doi: 10.1007/BF02392813 , ISSN   0001-5962, MR   0219666, S2CID   125806386
2. V. Lafforgue (2002), "K-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes" (PDF), Inventiones mathematicae , doi: 10.1007/s002220200213
3. Helgason (2010), "Integral Geometry and Radon Transforms", Springer , doi: 10.1007/978-1-4419-6055-9_9