We will illustrate some of the things discussed in the outline in some elementary examples. We will focus on the example GL(n), the group of n by n invertible matrices. In this example, many of the associated objects can be described explicitly in terms of matrices and linear algebra with a minimum amount of background knowledge.
For G=GL(n), we will describe Borel subgroups, a maximal torus, semisimple and unipotent elements, root subgroups, the Lie algebra, root spaces, the adjoint action, topological properties of GL(n, C) and GL(n R), number of points of GL(n, F_p). We will describe the characters, cocharacters, roots and coroots of GL(n) relative to the diagonal maximal torus; this will give us an example of a root datum for GL(n).
If we have time, we can discuss some representations of GL(n) as well, such as the symmetric and exterior powers.
Much of the rest of the course will focus on explaining how these objects make sense for a general class of groups, of which GL(n) is the prototypical example. The class is that of connected reductive groups, notions that will be defined later in the course.