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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 $\mathcal{S}(M)$?

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    $\begingroup$ +1 because I have been wondering about getting some kind of algebra of Schwartz functions on a unimodular Lie group -- there are notions for certain Lie groups, due originally (I think) to Harish-Chandra and elaborated by others, but I am a complete novice in that area and don't know if those definitions only make sense for e.g. radial functions. $\endgroup$
    – Yemon Choi
    Commented Nov 5, 2011 at 0:00
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    $\begingroup$ Regarding your last question: if your group is nonabelian then the Fourier transform ought to naturally take values in some topological algebra of operators on some direct sum of "well-behaved irreducible representations". E.g. if your group is compact then there is a well-developed notion of the nonabelian Fourier transform from $L^1(G)$ into $\bigoplus_\pi {\rm End}(H_\pi)$. $\endgroup$
    – Yemon Choi
    Commented Nov 5, 2011 at 0:03
  • $\begingroup$ @Yemon Choi: For closed manifolds, can we use the Nash embedding theorem to treat it some kind of submanifolds in Euclidean space? $\endgroup$
    – Bombyx mori
    Commented Dec 11, 2014 at 3:25

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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

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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).

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There was a whole "mini thesis" devoted to this question - "Schwartz functions on Nash manifolds" by A. Aizenbud and D. Gourevitch

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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).

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