Representations of algebraic groups over a 2-dimensional local field

We introduce a categorical framework for the study of representations of $G_F$, where $G$ is a reductive group, and $\text{bF}$ is a 2-dimensional local field, i.e. $F=K((t))$, where $K$ is a local field. Our main result says that the space of functions on $G_F$, which is an object of a suitable category of representations of $G_F$ with the respect to the action of $G_F$ on itself by left translations, becomes a representation of a certain central extension of $G_F$, when we consider the action by right translations.