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?

unitaryrepresentations, so one can't get decomposition of the left-reg rep of SL(2,C) as a sum of fin-dim reps. But your question is slightly different. Nonetheless, SL(n,k) or its universal cover seems an obvious first case to consider – Yemon Choi Oct 30 '09 at 22:24`SL(2)`

algebraic`<--->`

`SU(2)`

complex (for the purposes of this question at least). – Ilya Nikokoshev Oct 30 '09 at 23:04