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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 Schwarz Schwartz space is in Casselman's paper "Introduction to the Schwarz 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).
For Lie groups, at least for those that embed into $GL_n(R)$ for some $n$, my favorite treatment of the Schwarz space is in Casselman's paper "Introduction to the Schwarz 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).