Since Springer’s construction of representations of Weyl groups in 1978, there has been an explosion of constructions of groups and algebras (e.g., quantum groups) acting on homology groups of spaces. This is very different from older constructions such as the Borel-Weil theorem, in which the Lie group acts on the space and a coherent sheaf, hence on its sheaf cohomology; in “geometric representation theory” the group or algebra action is only on the homology (topological, not really algebro-geometric).
One of the many benefits of this approach is that the representations generally come with canonical bases and inner products, though they may be hard to calculate.
I will assume some rudiments of the representation theory of Lie groups and algebraic topology, mainly homology. We will probably need some K-theory, intersection homology, and maybe D-modules, all of which will be developed as needed. To the extent that we follow a textbook, it will be Chriss & Ginzburg’s Complex geometry and representation theory.