# The $p$-adic groups and their Weyl groups

## Meeting Details

Abstract: In a 1957 paper, Tits explained the analogy between the symmetric group $S_n$ and the general linear group over a finite field $\mathbb F_q$ and indicated that $S_n$ should be regarded as the general linear group over $\mathbb F_1$, the field of one element. Following Tits' philosophy, we would like to regard the affine Weyl groups as the reductive group over $\mathbb Q_1$, 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.