Reference request for algebraic Peter-Weyl theorem?

Solution 1:

This will be true for any complex reductive group. A general Frobenius reciprocity argument shows that $\mathrm{Hom}_G(V,\mathbb C[G]) \cong V^{\vee}$ as $G$-representations. On the other hand, since $G$ is reductive, $\mathbb C[G]$ is a direct sum of irreducible reps. Putting these two observations together proves that indeed $\mathbb C[G] \cong \bigoplus_{V \text{ irred.} } V\boxtimes V^{\vee}$.

It is a good exercise to check this concretely when e.g. $G = \mathrm{SL}_2$.

Solution 2:

There is a proof of this statement for any complex reductive group, more or less along the lines of Matt's sketch, in Chapter 12 of Goodman and Wallach's book "Representations and invariants of the classical groups".