As a unimodular subgroup of the group of automorphisms of $\mathbb{R}^2$, $\operatorname{SL}(2,\mathbb{R})$ can be represented as a subgroup of $\mathcal{U}(L^2(\mathbb{R}^2))$ (the group of unitary operators on $L^2(\mathbb{R}^2)$), where each element $A \in \operatorname{SL}(2,\mathbb{R})$ is mapped to a unitary operator $R_A$ defined by $$ R_A \xi(X):= \xi(A^{-1}X) $$ for all $\xi \in L^2(\mathbb{R}^2)$ and $X \in \mathbb{R}^2$. Applying the Fourier transform on $L^2(\mathbb{R}^2)$, it seems to me that this representation has to be irreducible. Is this true?
If that is so, to which irreducible unitary representation of $\operatorname{SL}(2,\mathbb{R})$, is it equivalent? (I found the list of irreducible unitary representations of $\operatorname{SL}(2,\mathbb{R})$ in Lang's book.)