The Schwartz Space on a Manifold I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here.
This question is also vaguely related (both questions arose from the same thing I was working on) to this question I just asked last night.
The question is simple:  on a general manifold $M$, can one generalize the space of Schwartz functions on $\mathbb{R}^n$ to a space of smooth functions $\mathcal{S}(M)$ on $M$ that obeys similar properties?  I would like to be able to define the convolution of two Schwartz functions, so I guess I better require that $M$ at least be a (unimodular) Lie group.  Is it then possible to define $\mathcal{S}(M)$?  What about the Fourier transform?  Is there a natural definition of the Fourier Transform on $\mathcal{S}(M)$?
 A: To define a Schwartz space, you need a notion of decay at infinity, so you need a ``norm'', i.e. a distance to some origin. So the convenient framework is a complete Riemannian manifold. However, even on a Lie group, it is not enough to choose an invariant Riemannian structure to get a Schwartz space having the properties that you require (convolution algebra, good Fourier transformation...). See e.g. the subtlety in the definition of Harish-Chandra's Schwartz space on a semi-simple Lie group, where you have to throw in the $\Xi$-function.
For simply connected solvable Lie groups, the definition of the Schwartz algebra is (I believe) fairly recent: see a paper by Emilie David-Guillou: https://arxiv.org/pdf/1002.2185
A: For Lie groups, at least for those that embed into $GL_n(R)$ for some $n$, my favorite treatment of the Schwartz space is in Casselman's paper "Introduction to the Schwartz Space of $\Gamma \backslash G$", Can. J. Math. XL, No 2, 1989.  There Casselman defines an appropriate Schwarz space on $\Gamma \backslash G$ whenever $G$ is the Lie group obtained by taking the $R$-points of an affine algebraic group over $R$, and $\Gamma$ is any discrete subgroup of $G$ (including the trivial subgroup).  
I think this is the right place to look, before studying things like the Fourier transform (i.e. Plancherel and Paley-Wiener theorems).
A: There was a whole "mini thesis" devoted to this question - "Schwartz functions on Nash manifolds" by A. Aizenbud and D. Gourevitch
A: May be this paper (cited in the relevant paper of A. Aizenbud and D. Gourevitch mentioned by  George) is also relevant for your problem :
du Cloux, Fokko: Sur les repr\'esentations diff\'erentiables des groupes de Lie
 alg\'ebriques. Annales scientifiques de l E.N.S  tome 24, no 3, p. 257-318
(1991).
