Algebraic Peter-Weyl theorem in the case of $G=SL_2$. The algebraic Peter-Weyl theorem says that for a linear reductive group $G$ we have $\mathbb{C}[G] = \oplus_{V} V \otimes V^* $, where $V$ runs over the set of all non-isomorphic irreducible representations of $G$. 
I would like to know this result explicitly in the case of $G=SL_2$. We have $\mathbb{C}[SL_2] = \mathbb{C}[a,b,c,d]/(ad-bc-1)$. How to write $\mathbb{C}[SL_2] = \oplus_{V} V \otimes V^* $ explicitly (write the representations $V$ explicitly)? Thank you very much.

The function $V \otimes V^* \to \mathbb{C}[G]$ is given by $(v, \lambda) \mapsto (g \mapsto \lambda(gv))$. Let $G = SL_2$ and write the coordinate on $SL_2$ as $$ \left( \begin{matrix} a & b \\ c & d \end{matrix} \right). $$
As shown in the post, $Span_{\mathbb{C}}\{c, d\}$ is a representation of $G$. Let $V = Span_{\mathbb{C}}\{c, d\}$ and take $e_1 = c = (1, 0)^T$, $e_2 = d = (0, 1)^T$. Then $V^* = Span_{\mathbb{C}}\{e_1^*, e_2^*\}$, where $e_i^*(e_j)=\delta_{ij}$. Let $v = e_1$, $\lambda=e_1^*$. Then $(e_1, e_1^*) \mapsto (g \mapsto e_1^*(ge_1))$. Suppose that $$ g = \left( \begin{matrix} g_{11} & g_{12} \\ g_{21} & g_{22} \end{matrix} \right). $$ $e_1^*(ge_1) = e_1^*(g_{11}, g_{21})^T = e_1^*(g_{11} e_1 + g_{21} e_2) = g_{11}$. Therefore $(e_1, e_1^*) \mapsto x_{11}$, where $x_{11}$ is the function on $SL_2$ whose value on a matrix $g$ in $SL_2$ is the $(1,1)$ entry of $g$. Similarly $(e_i, e_j^*) \mapsto x_{ji}$. We have $x_{11}x_{22} - x_{12} x_{21} = 1$ since $x_{ij}$'s are functions on $SL_2$.
