I am looking for a classification of irreducible real representations of $\mathrm{SL}(2,\mathbb{R})$ of finite dimension (in the following by "representation" I mean a representation of finite dimension). 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 $a \geq 0$?
Best Answer
First, the classification of complex representations of $\mathrm{SL}_2(\mathbb{R})$ is the same as that of $\mathrm{SL}_2(\mathbb{C})$. This is because any representation of the Lie group $\mathrm{SL}_2(\mathbb{R})$ gives a representation of the Lie algebra $\mathfrak{sl}_2(\mathbb{R})$. But by the universal property of complexification any such representation factors through $\mathfrak{sl}_2(\mathbb{C})$. A priori these representations may lift only to the universal cover, but since all these representations factor through the map $\widetilde{\mathrm{SL}_2(\mathbb{R})} \rightarrow \mathrm{SL}_2(\mathbb{C})$ they factor through $\mathrm{SL}_2(\mathbb{R})$.
Now, we still need to sort out the real forms of these representations, that is we need to sort out whether the irreps support an invariant symmetric bilinear form, an antisymmetric bilnear form, or are not self-dual. In particular, it's enough to just observe that every representation has a real form. But $\mathrm{Sym}^n(\mathbb{R}^2)$ is a direct construction of a real form of each irrep.
So the classification is exactly the same as in the complex case.