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 Xx(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.