I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R}).$ There is a complete classification of complex representations of $\mathrm{SL}(2,\mathbb{C}).$ More precisely, if $V$ is an irreducible complex representation of $\mathrm{SL}(2,\mathbb{C})$, then $V$ is isomorphic to $\mathrm{Sym}^a(\mathbb{C}^2)$ for some $a \geq 0.$

Is there a similar classification in the case of real representations of $\mathrm{SL}(2,\mathbb{R})$? Is any irreducible real representation of $\mathrm{SL}(2,\mathbb{R})$ isomorphic to $\mathrm{Sym}^a(\mathbb{R}^2)$ for some $a \geq 0$?