A linear algebraic group `G` defined over a field `F` is, by definition, a group object in the category of affine `F`-varieties. As such, `G` can be studied using methods from algebraic geometry and group theory.
If `F` is a local field, e.g. the `p`-adic numbers, the geometry of `G` has number-theoretic significance. Such groups are used frequently in the Langlands program, a very influential research direction linking number theory with representation theory.

My main object of interest are affine Deligne-Lusztig varieties. These are finite-dimensional schemes over the residue field of `F`, and are used primarily to study the special fibre of Rapoport-Zink moduli spaces. Via these Rapoport-Zink spaces, the geometry of affine Deligne-Lusztig varieties is related to Shimura varieties.

The definition of affine Deligne-Lusztig varieties is motivated by a classical paper of Deligne and Lusztig. In a non-affine setting, they construct certain varieties, whose cohomology provides important representations of finite groups of Lie type. The geometry of these Deligne-Lusztig varieties is well-understood. This connection is not only motivational – sometimes the nice geometric properties of classical Deligne-Lusztig varieties come up when studying the affine case.

On a more down-to-earth level, affine Deligne-Lusztig varieties compare two decompositions of the space `G(F˘)`. The first one is the decomposition into σ-conjugacy classes, a Galois twisted version of the usual conjugation in a group. The second one is the decomposition into double cosets `IxI`, where `I` is an Iwahori subgroup and `x∈G(F˘)`. The intersection of such a σ-conjugacy class `[b]` with an Iwahori double coset `IxI` is called Newton stratum, and its geometry is closely related to a corresponding affine Deligne-Lusztig variety `X _{x}(b)`. Both decompositions are can be parametrized using an infinite Coxeter group, known as the extended Weyl group of

`G`.

The tools to study these affine Deligne-Lusztig varieties come from algebraic geometry, the theory of Coxeter groups, the theory of (affine) Lie algebras and (affine) root systems as well as quantum algebra. This makes affine Deligne-Lusztig varieties a very rich and dynamic research topic.