There's a well-known decomposition of $L^2(G)$, a regular representation of compact complex group Lie $G$, called **[Peter-Weyl theorem](http://en.wikipedia.org/wiki/Peter%E2%80%93Weyl_theorem)**.


Turns out for some reason I [automatically think](https://mathoverflow.net/questions/3446/tannakian-formalism/3466#3466) that there is a similar theorem that decomposes regular representation $k[G]$ of *algebraic* group $G$:


$$k[G] = \bigoplus\_R \ R^\* \otimes R$$


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 $\mathbb G\_m$. Then $k[\mathbb G\_m] = k[x, x^{-1}]$ where each summand $k\cdot x^n$ is a separate representation of $\mathbb G\_m$ into $\mathbb G\_m = GL(1, k)$, 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?



Decomposition of k[G]
This is true for reductive groups, more or less by definition. An algebraic representation of an algebraic group is a comodule V over the algebra of functions O(G) of the group. Therefore, every representation V induces a map
V -> V ⊗ O(G), or equivalently V^\* ⊗ V --> O(G) (call the source of this map C(V) for coefficient space of V). It is not hard to see that the latter is a map of G x G modules. If G is reductive, then its representation category is semi-simple, and thus so is the representation category of G x G. In this case the simples of G x G are external tensor product of simples of V, and Hom(A ⊗' B, C ⊗' D) = d(A,C) ⊗ d(B,D) where d(V,W)=0 if v \cong W, C else. Here ⊗' means external tensor product. There doesn't appear to be a ⊠


For non-reductive groups, you can still form O(G) in an analogous way:


Let A = ⊕V V^\* ⊗' V, where here the sum is over ALL finite dimensional modules V (not just isoclass representatives, and not just simples), and again the tensor product is external, so this lives in a completion of Rep(G) ⊗' Rep(G), and ⊗' means Deligne tensor product of categories.


Well this A is way too big, but now let's quotient A by the images of f^\* ⊗' id - id ⊗' f, for all f:V-->W. This cuts A back down, for instance it identifies C(V) and C(V') whenever V and V' are isomorphic. If the category Rep(G) is semi-simple, you can similarly use the projectors and inclusions of simple objects to reduce to a Peter-Weyl type decomposition.


One nice thing about this construction (even in the semi-simple case) is that it is basis free because you don't choose representatives of simple objects, and also it makes the multiplication structure completely trivial:
V^\* ⊗' V ⊗2 W^\* ⊗' W = V^\* ⊗ W^\* ⊗' V ⊗ W --> W^\* ⊗ V^\* ⊗' V ⊗ W,
using the braiding (tensor swap). It also works in braided tensor categories and explains the multiplication structure on the "covariantized" quantum group.


