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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...

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    Yes! It looks like what I was looking for is the theorem of section 3.1 in Chapter 7. It looks like Matt E's argument below with a by-hand argument that $V^* \boxtimes V$ is the $V$-isotypic component of $\mathbb C[G]$ instead of the general Frobenius reciprocity statement. Thanks!2011-10-23

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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. $G = \mathrm{SL}_2$.

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    @MattE, I think that in $G=SL_2$ case, we have $\mathbb{C}[G] = V_{0} \oplus V_{\omega_1} \oplus V_{\omega_1}^{\vee} \oplus \cdots$, where $V_0 = \mathbb{C}$, $V_{\omega_1}$ is the two-dimensional vector space with a basis $\{b_{11}, b_{21}\}$, $V_{\omega_1}^{\vee}$ is the two-dimensional vector space with a basis $\{b_{11}, b_{12}\}$. I think that there are some other $V_{\lambda}$'s on the right hand side since $\mathbb{C}[SL_2]$ is an infinite dimensional vector space. But I don't know what are these $V_{\lambda}$'s. Do you know what are these $V_{\lambda}$'s? Thank you very much.2015-08-02
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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".

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    @sibilant, do you know how to verify $\mathbb{C}[G] = \oplus_V V^* \boxtimes V$ in the case of $G=SL_2$? Thank you very much.2016-05-04
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A good refernce in my opinion is Theorem 27.3.9 in the book by Patrice Tauvel and Rupert Yu named Lie Algebras and Algebraic Groups.