Harish-Chandra Schwartz space - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive Lie group in the Harish-Chandra class, choose a maximal compact subgroup $K$ , and let $\sigma:G\to[0,\infty)$ be a standard proper $K$ -bi-invariant length function. The Harish-Chandra Schwartz space $\mathcal C(G)$ is the space of smooth functions $f:G\to\mathbb C$ such that, for every $D_1,D_2$ in the universal enveloping algebra of the complexified Lie algebra and every integer $N\geq0$ ,

\sup_{g\in G}(1+\sigma(g))^N\Xi(g)^{-1} \lvert (L_{D_1}R_{D_2}f)(g)\rvert<\infty.

Here $\Xi$ is the Harish-Chandra Xi-function, and $L_{D_1}$ , $R_{D_2}$ are the corresponding left- and right-invariant differential operators.

Topological algebra structure

The displayed seminorms make $\mathcal C(G)$ a Fréchet space, independently of the standard choices up to equivalent seminorms. It is closed under convolution and under the involution $f^*(g)=\overline{f(g^{-1})}$ , because real reductive groups are unimodular. With these operations it is a Fréchet $*$ -algebra and a basic test algebra for tempered harmonic analysis on $G$ Knapp, Chapter VII.

Comparison with ordinary rapid decay

On an abelian vector group, ordinary Schwartz seminorms use polynomial weights and constant-coefficient derivatives. On a noncompact reductive group, Haar volume grows exponentially. The factor $\Xi(g)^{-1}$ compensates for the characteristic decay of spherical matrix coefficients, while $(1+\sigma(g))^N$ enforces additional rapid polynomial decay. Omitting $\Xi^{-1}$ therefore gives the wrong scale for harmonic analysis on $G$ .

Conventions and scope

Some sources define $L_{D_1}R_{D_2}$ with an antipode or reverse-order convention; these choices yield the same space. The notation $\mathcal C(G)$ is conventional but can be confused with continuous functions. This definition concerns smooth scalar-valued functions on a real reductive group, not the Schwartz–Bruhat space of an arbitrary locally compact abelian group.

References

Anthony W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton University Press, 1986. Author-maintained record. Relevant: Chapter VII on spherical functions, estimates, and the Harish-Chandra Schwartz space.
Harish-Chandra, “Harmonic analysis on real reductive groups I: The theory of the constant term,” Journal of Functional Analysis 19 (1975), 104–204. DOI record. Relevant: the Schwartz space and constant-term theory.