A lot of the structure of a compact Lie group can be understood via algebraic objects of a discrete nature, such as the character lattice of a maximal embedded torus. Amongst other things, this allows us to explicitly describe its representation theory.
The talk will be about the circle of ideas surrounding Weyl’s computation of the characters of irreducible modules in terms of the highest weight. I’ll then sketch some further developments of the theory, such as the Borel-Weil geometric realization of these representations via line bundles on flag varieties.