Abstract:
We will describe the local unramified Langlands correspondence for two-dimensional local fields (following an approach of M. Kapranov). We will construct the categorical analog of principal series representations of general linear groups over two-dimensional local fields, describe the properties of the construction, and discuss some hypothesis. The main ingredients of the construction are some central extensions of these groups (these groups are defined over two-dimensional local fields or over adelic rings of two-dimensional arithmetic schemes). We will prove reciprocity laws for these central extensions, i.e., splittings of these central extensions over some subgroups defined over semilocal rings constructed by means of points and one-dimensional subschemes of a two-dimensional arithmetic scheme.
