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?
