= Smooth automorphic form
A <smooth automorphic form> is an <automorphic form> whose archimedean variables are infinitely differentiable, without necessarily being <holomorphic>. Conventions differ over whether compact-group finiteness is included or one takes a larger smooth representation space. In the classical finite-vector convention on $G=\operatorname{SL}_2(\mathbb R)$, one asks for left invariance under an arithmetic subgroup $\Gamma$, moderate growth, finite-dimensional span under right translations by $K=\operatorname{SO}(2)$, and finiteness under the centre of the <universal enveloping algebra>.<Classical Modular Forms as Automorphic Forms by Matthew Emerton>[Definition 5.1 and section 7, general forms versus holomorphic forms]{ref}
The constant <function> on $\Gamma\backslash G$ is an example: it is smooth, bounded, $K$-invariant and annihilated by positive-order invariant differentiation. More interesting examples come from <Maass forms>, whose <Laplacian> eigenfunction condition replaces holomorphy. Smoothness is therefore a regularity requirement, not a claim that the <Cauchy-Riemann equations> hold.
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