Automorphic form

Remarks

Example: let $G = \mathrm{SL}_2(\R)$, $K = \mathrm{SO}(2)$, $\Gamma = \mathrm{SL}_2(\Z)$, and let $h$ be a modular form of weight $k$. For $g = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ put $f(g) = h(g \cdot i)\,(ci + d)^{-k}$, where $g \cdot i = \frac{ai + b}{ci + d}$. The modularity of $h$ gives clause 1. Right translation by the rotation $k_\theta = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ multiplies $f$ by $e^{-ik\theta}$, since $k_\theta \cdot i = i$ and $i \sin\theta + \cos\theta = e^{i\theta}$, which gives clause 2. Holomorphy of $h$ makes the Casimir operator act on $f$ by a scalar, which gives clause 3. The bound of $h$ at the cusp gives clause 4.

Non-example: on the same $G$, the function $f(g) = (\mathrm{Im}\, g \cdot i)^s$ satisfies clauses 2, 3 and 4, but it is not invariant under $\gamma = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \in \Gamma$, so clause 1 fails.

This is the definition of Harish-Chandra. A classical variant works on a symmetric space $G/K$ and allows a factor of automorphy $j(\gamma, z)$ in clause 1, as for modular forms of weight $k$. In the adelic setting, $G$ is a reductive group over a number field $F$ and automorphic forms are functions on $G(F) \backslash G(\mathbb{A}_F)$.