Delooping and reciprocity

The Contou-Carrère symbol has been introduced in the 90's in the study of local analogues of autoduality of Jacobians of smooth projective curves. It is closely related to the tame symbol, the residue pairing, and the canonical central extension of loop groups. In this talk we will a discuss a K-theoretic interpretation of the Contou-Carrère symbol, which allows us to generalize this one-dimensional picture to higher dimensions. This will be achieved by studying the K-theory of Tate objects, giving rise to natural central extensions of higher loop groups by spectra. Using the K-theoretic viewpoint, we then go on to prove a reciprocity law for higher-dimensional Contou-Carrère symbols. This is joint work with O. Braunling and J. Wolfson.