It seems that, for $GL_n$, and possibly for something like complex reductive groups $G$ in general, there's an algebraic version of the Peter-Weyl theorem, which might say that the coordinate ring of $G$ decomposes as a direct sum of endomorphisms of all the irreducible algebraic representations. That is, that
$$\mathbb C[G] = \bigoplus V^* \boxtimes V$$
as $G \times G$-representations (sum is over all irreps of $G$).
Does anyone happen to know of a reference for this such an algebraic kind of Peter-Weyl?
Grounds for suspicions: this blog post by David Speyer, this comment by Ben Wieland on mathoverflow, and the first few lines of notes from lecture 5b of an MIT seminar on quantum groups that one can find by searching for "peter-weyl algebraic" and that I'm not allowed to link to.
A clean and true statement for when such a theorem might hold would be a lovely start as well, I suppose…
Best Answer
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. $G = \mathrm{SL}_2$.