Analog of Peter-Weyl theorem for $G[[t]]$ - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:28:33Z http://mathoverflow.net/feeds/question/68626 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68626/analog-of-peter-weyl-theorem-for-gt Analog of Peter-Weyl theorem for $G[[t]]$ Alexander Braverman 2011-06-23T16:28:22Z 2011-06-23T16:28:22Z <p>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?</p>