Harish-Chandra Schwartz algebra - Imported Knowlpedia Content

Definition

Let $G$ be a real reductive Lie group in the Harish-Chandra class. The Harish-Chandra Schwartz algebra is the Harish-Chandra Schwartz space $\mathcal C(G)$ , equipped with convolution

(f*h)(x)=\int_G f(y)h(y^{-1}x)\,dy

and the involution $f^*(x)=\overline{f(x^{-1})}$ . Its locally convex topology is generated by the rapid-decay seminorms involving left and right invariant derivatives, polynomial weights, and the Harish-Chandra Xi-function. With these data, $\mathcal C(G)$ is a complete Fréchet $*$ -algebra: multiplication and involution are continuous for its defining topology.

Analytic structure

The left-convolution map, initially defined for compactly supported smooth functions, extends to $\mathcal C(G)$ and realizes it as a dense involutive subalgebra of the reduced group $C^*$ -algebra $C_r^*(G)$ . The space is also dense in $L^2(G)$ . Its stronger Fréchet topology retains all invariant derivatives and rapid-decay estimates, so convolution can be differentiated and estimated without leaving the space. This is the standard $\mathcal C^2(G)$ scale; it should not be confused with the $L^1$ -Schwartz space $\mathcal C^1(G)$ , whose estimates use a different power of $\Xi$ . Closure under convolution is a substantive theorem of Harish-Chandra's Schwartz-space theory Harish-Chandra, Schwartz-space construction.

Representation-theoretic role

Every tempered unitary representation acts on $\mathcal C(G)$ through its integrated form. The algebra therefore provides a common test-function domain for tempered characters, Plancherel decomposition, and harmonic analysis, while its inclusion in the reduced group $C^*$ -algebra connects smooth representation theory with operator-algebraic completion.

Spherical subalgebra

After choosing a maximal compact subgroup $K$ , the $K$ -bi-invariant part $\mathcal C(G//K)$ is commutative. The spherical transform identifies it with a Weyl-invariant Schwartz-type function algebra on the spectral parameter space; this is the content of the Trombi–Varadarajan theorem Trombi–Varadarajan, main theorem.

References

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 construction and convolution estimates.
P. C. Trombi and V. S. Varadarajan, “Spherical Transforms on Semisimple Lie Groups,” Annals of Mathematics 94 (1971), 246–303. DOI record. Relevant: the spherical transform of Harish-Chandra Schwartz spaces.