Consider the Lie-group $\text{SL}_2(\mathbb R)$ with the canonical basis $X,W,H$. Element of this basis act on distributions on $SL_2(\mathbb{R})$ by the lie-derivatives $\mathcal{L}_X,\mathcal{L}_H,\mathcal{L}_W$. Suppose $\mathcal{L}_Xf=0,\mathcal{L}_H=\lambda f$, find the Fourier expansion of $f$.
I know that there are two possible solutions ($\delta_0$ and another one) but how can I distinguish between them?