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There's a well-known decomposition of L2(G), $L^2(G)$, a regular representation of compact complex group Lie G, $G$, called Peter-Weyl theorem.

Turns out for some reason I automatically think that there is a similar theorem that decomposes regular representation k[G]$k[G]$ of algebraic group G:

               k[G] $G$:



$$k[G] = \osum bigoplus_R \ R^* \otimes R
R$$



where sum goes over representations to GL(n, k)$GL(n, k)$. For this to work I think we need G $G$ to be a linear reductive group over, say, algebraically closed field k$k$ of characteristic 0. Also, perhaps we need pi_1(G) $\pi_1(G) = 11$. 



But perhaps this is not true — the search hasn't given me a reference yet, but I wasn't able to provide a counterexample either.



Consider, for example, the multiplicative group G_m$\mathbb G_m$. Then k[G_m$k[\mathbb G_m] = k[x, x^{-1}]x^{-1}]$ where each summand k x^n$k\cdot x^n$ is a separate representation of G_m$\mathbb G_m$ into $\mathbb G_m = GL(1, k)k)$, specifically the one given by $a \mapsto a^na^n$. So the identity works.



So, is there such a theorem? What's a reference or a counterexample?
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Algebraic Peter-Weyl theoremDecomposition of k[G]

There's a well-known decomposition of L2(G), a regular representation of compact complex group Lie G, called Peter-Weyl theorem.

Turns out for some reason I automatically think that there is a similar theorem that decomposes regular representation k[G] of algebraic group G:

               k[G] = \osum R^* \otimes R

where sum goes over representations to GL(n, k). For this to work I think we need G to be a linear reductive group over, say, algebraically closed field k of characteristic 0. Also, perhaps we need pi_1(G) = 1.

But perhaps this is not true — the search hasn't given me a reference yet, but I wasn't able to provide a counterexample either.

Consider, for example, the multiplicative group G_m. Then k[G_m] = k[x, x^{-1}] where each summand k x^n is a separate representation of G_m into G_m = GL(1, k), specifically the one given by a \mapsto a^n. So the identity works.

So, is there such a theorem? What's a reference or a counterexample?
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There's a well-known decomposition of L2(G), a regular representation of compact complex group Lie G, called Peter-Weyl theorem.

Turns out for some reason I automatically think that there is a similar theorem that decomposes regular representation k[G] of algebraic group G:

               k[G] = \sum osum R^* \otimes R

where sum goes over representations to GL(n, k). For this to work I think we need G to be a linear reductive group over, say, algebraically closed field k of characteristic 0. Also, perhaps we need pi_1(G) = 1.

But perhaps this is not true — the search hasn't given me a reference yet, but I wasn't able to provide a counterexample either.

So, is there such a theorem? What's a reference or a counterexample?
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