Relationship between Harish-Chandra Schwartz space and more generic Schwartz spaces

Question

If $G$ is a connected semisimple Lie group with finite center, Harish-Chandra defined a Schwartz space of rapidly decreasing functions on $G$ as the space of $\mathrm{C}^\infty$ functions defined by the seminorms

$$\|f\|_{D_1, D_2, r} = \sup_{g \in G} \frac{(1 + \sigma(g))^r |f(D_1 ; g ; D_2)|}{\Xi(g)}$$

where $D_1$ and $D_2$ are left and right-invariant differential operators on $G$ respectively, $r \in \mathbb{R}^+$, $D_1 ; g ; D_2$ is the two-sided action of $D_1$ and $D_2$ on $g$, and $\sigma$ and $\Xi$ are the standard functions defined by Harish-Chandra.

An arguably more intuitive definition of rapidly decreasing function that matches the use in other contexts would be the space of $f \in \mathrm{C}^\infty$ defined by the seminorms

$$\|f\|_P = \sup_{g \in G} |P f(g)|$$

for every polynomial $P$ in one-sided-invariant differential operators on $G$.

What is the relationship between these two definitions of rapidly decreasing function? Why is it necessary to use a more particular definition of Schwartz space for semisimple Lie groups? There doesn't appear to be anything stopping one from turning the latter definition into a usable convolution algebra.

Answer 1

Belatedly...

I think this is a very reasonable question, if only for the implicit question "what would characterize a 'correct' Schwartz space?"

Yes, from one side, as in the question, something about rapid decay of derivatives... In Harish-Chandra's presentation, there is a specific gauge of "rate of growth", with functions naturally arising from the repn theory.

And, then, since H-C proved several theorems about things that were harmonious with his defn of "Schwartz" functions, in that regard his definition is "correct".

But, still, a feature that is a bit fancier, but is very useful, is that a "good" notion of "Schwartz space" on a smooth manifold should include its nuclearity, meaning (for me) that it admits tensor products with similar TVS's. The simplest case of "nuclearity" is as a countable (projective) limit of Hilbert spaces, with Hilbert-Schmidt transition maps. Pairs of such things admit genuine-categorical tensor products! Explicitly, $$ (\lim_n X_n) \otimes_{categorical} (\lim_n Y_n) =\lim_n (X_n\otimes_{HS} Y_n) $$ where $\otimes_{HS}$ is the Hilbert-Schmidt norm completion of the algebraic tensor product. The existence of a genuine tensor product gives the Hom-tensor adjunction, also known as the Cartan-Eilenberg adjunction, which gives a one-line proof of (most of) a Schwartz Kernel Theorem, for example.

I myself have not carried out any verification of the nuclearity of H-C's "Schwartz space". Casselman's notes on such things explicitly disaver such verification, also. Possibly someone has done this, or can give a pointer, but I cannot, even after a modest effort. :)