Let $G$ be a reductive group over ${\mathbb C}$ and let $G[[t]]$ denote the corresponding group over the formal power series ring ${\mathbb C}[[t]]$. This is a group scheme, so one can speak about its ring of functions (by definition this is the direct limit of functions on $G({\mathbb C}[[t]]/t^n)$). Can one say anything about this algebra as a representation of $G[[t]]\times G[[t]]\rtimes {\mathbb C}^* $ or even as a representation of $G\times G\times {\mathbb C}^* $ (${\mathbb C}^*$ acts by rotating $t$)? For example, if $G=SL(2)$ and $V_n$ is the space of polynomials of degree $n$ on ${\mathbb C}^2[[t]]$ (naturally a representation of $G[[t]]\rtimes {\mathbb C}^*$), then can one describe the above ring of functions as the direct sum of $V_n\otimes V_n$ with some multiplicity space?
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Why are you defining the ring of functions to be a direct limit? For example, when $G = GL_1$, the ring of functions under the usual definition is $\mathbb{C}[[t]]^\times$, which does not look like a direct limit of $(\mathbb{C}[t]/(t^n))^\times$. Are you working formally in a category of pro-objects that in this case happen to be representable by affine schemes? – S. Carnahan Jun 24 '11 at 07:31
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I don't see any other definition (what do you mean by ${\mathbb C}[[t]]^{\times}$ being the ring of functions? It is not a ring...) – Alexander Braverman Jun 25 '11 at 11:22
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1Well, I agree that your definition of "ring of functions" looks slightly convoluted. Your $G[[t]]$ is obtained by base change from $G$, which is affine, so the ring of functions of $G[[t]]$ is simply the ring of functions of $G$ tensor $\mathbb{C}[[t]]$ (by definition of base change or product of schemes). In the example of $GL_1$, the ring of functions (over $\mathbb{C}$) is $\mathbb{C}[u]$ localized at $u$ (the determinant is invertible), in other words the ring of Laurent polynomials $\mathbb{C}[u,u^{-1}]$, and the ring of functions of $G[[t]]$ is $\mathbb{C}[[t]][u,u^{-1}]$. – Baptiste Calmès Jun 28 '11 at 09:10