Abstract: In a 1957 paper, Tits explained the analogy between the symmetric group S_n and the general linear group over a finite field F_q and indicated that S_n should be regarded as the general linear group over F_1, the field of one element.
Following Tits' philosophy, we would like to regard the affine Weyl groups as the reductive group over Q_1, the 1-adic field. Although it is still premature to develop the theory of 1-adic field at the current stage, we do have a fairly good understanding on the conjugacy classes of the affine Weyl groups, together with the length function on it, and such knowledge allows us to reveal a great part of the structure of the conjugacy classes of p-adic groups. I will explain how this idea may be used in representation theory and in arithmetic geometry.

Xuhua He is currently a professor of mathematics at the University of Maryland. Professor He is a leading figure in the field of representation theory. He made fundamental contributions to various aspects of arithmetic geometry and representation theory, especially affine Deligne-Lusztig varieties. Professor He was an invited speaker of ICM2018. He received a Morningside Gold medal in 2013.