The hyperbolic Laplacian of weight zero is defined as $$\Delta = - y^2 (\partial_x^2 + \partial_y^2),$$ which can be derived from the Riemannian metric on the hyperbolic upper half plane. However, there is the notion of a weight $k$ Laplacian, which is $$\Delta_k = - y^2 (\partial_x^2 + \partial_y^2) + i k y (\partial_x + i \partial_y).$$ The weight $k$ Laplacian satisfies $$\Delta_k (f \vert_k \gamma) = (\Delta_k f) \vert_k \gamma, \; \mathrm{for \; all \;} \gamma \in \mathrm{SL}_2(\mathbf{Z}),$$ where $f(z) \vert_k \gamma = (c z + d)^{-k} f \left( \frac{az + b}{cz + d} \right)$ for any complex-valued function $f$ on the upper half plane and $\gamma = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.
What I'm wondering is how to actually derive the "right" weight $k$ Laplacian. It seems to me that it does not come from any Riemannian metric, but that it is rather defined by its commutativity with the $\vert_k$-operator. Is there any canonical derivation of the weight $k$ Laplacian or any explanation how to get there?