Summary: 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 \(\mathbb R\oplus\mathrm{SO}_0(1,2)\). 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 \(\mathcal E(X)\) of continuous and infinitely differentiable functions on \(X\) and also function spaces formed by functions with exponential growth on \(X\).