The Milnor $K$-theory and the Shintani cocycle

The goal of this article is to complete the unfinished construction (due to Glenn Stevens in an old preprint) of a certain Milnor $K$-group valued group cocycle for $GL_n(\mathbb{Q})$ where $n$ is a positive integer, which we call the Stevens cocycle. Moreover, we give a precise relationship between the Stevens cocycle and the Shintani cocycle, which encodes key informations on the zeta values of totally real fields of degree $n$, using the $\operatorname{dlog}$ map of $K$-theory and the Fourier transform of locally constant functions on $\mathbb{Q}^n$ with bounded support. Roughly speaking, the Stevens cocycle is a multiplicative version of the Shintani cocyle.