Invariant subspaces in some function spaces on the light cone in $\mathbb R^3$

For certain topological vector spaces of functions on the light cone $X$ in $\mathbb R^3$ we obtain a complete description of all the closed linear subspaces which are invariant with respect to the natural quasiregular representation of the group $\mathbb R\oplus\operatorname{SO}_0(1,2)$. In particular, we give a description of irreducible and indecomposable invariant subspaces. Among the function spaces we consider we include, in particular, the spaces $C(X)$ and $\mathscr E(X)$ of continuous and infinitely differentiable functions on $X$ and also function spaces formed by functions with exponential growth on $X$.
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