TITLE: Decomposition of k[G]
QUESTION [12 upvotes]: There's a well-known decomposition of $L^2(G)$, 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 $G$:
$$k[G] = \bigoplus_R \ R^* \otimes R$$
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?

REPLY [6 votes]: This statement is false in general for algebraic groups. It's true in characteristic 0, but it is not in general true in positive characteristic. Instead, one has a weaker statement in positive characteristic (cf Proposition 4.20 on page 213 in Jantzen's "Algebraic Groups"): 

Let $G$ be a reductive linear algebraic group over an algebraically closed field of positive characteristic $k$. Then $k[G]$ has an increasing filtration whose subquotients are of the form $H(\lambda) \otimes H(-w_0 \lambda)$, where $\lambda$ runs over the dominant weights for $G$ and the $H(\lambda)$ are the modules arising as global sections of line bundles on the flag variety of $G$ (the so-called costandard modules for $G$). 

Moreover, this is true when $k[G]$ is considered as a $G\times G$-module.
Note that unlike in characteristic 0, these modules $V$ are not in general irreducible. (It's worth noting that the category of modules over a reductive algebraic group is not in general a semisimple category — this is only true in characteristic 0).