One constructs an $n$-dimensional irreducible representation for $\mathfrak{sl}(2,\mathbb{R})$ as follows: $\rho(h) = \text{diag}(n-1,n-3,\ldots,-n+3,-n+1), \rho(e) = \text{uppdiag}(n-1,n-2,\ldots,1), \rho(f) = \text{lowdiag}(1,2,\ldots,n-1)$, where 'uppdiag' refers to the diagonal "one up" from the diagonal in 'diag', and 'lowdiag' defined similarly.
My question is: is this the only irreducible, $n$-dimensional representation for $\mathfrak{sl}(2,\mathbb{R})$? If so, why? Can one invoke the correspondence between Lie group and algebras for a simply connected lie group, along with the fact that $SL(2,\mathbb{R})$ has a unique $n$-dimensional representation? Could one somehow use the fact that $\mathfrak{sl}(2,\mathbb{C})$ has a unique irreducible $n$-dimensional representation?
A reference to the result would also be appreciated... I could only seem to find results on $SL(2,\mathbb{R})$ and $\mathfrak{sl}(2,\mathbb{C})$.
Thanks in advance!!