Consider homogeneous functions on $\mathbb{R}^2$, that is to say such that
$$f(ax,ay) = |a|^k f(x,y)$$
Then $GL_2(\mathbb{R})$ acts naturally on this space by $$g f(x,g) = f(g^{-1}(x,y)) |\det(g)|^{k/2}$$
Apparently, if it is an $SO(2)$ invariant function, it is an eigenvelue of the Laplace operator (on the hyperbolic place). I do not see why, can anyone help?