Representations of Lie Groups
MIT OpenCourseWare offers this advanced graduate course on representation theory of compact and non-compact Lie groups, building on 18.745 and 18.755 Lie Groups and Lie Algebras without requiring full mastery of them. Topics span continuous representations, K-finite and analytic vectors, admissible and unitary representations, the Harish-Chandra admissibility and analyticity theorems, (g,K)-modules, representations of SL(2,R), the Chevalley restriction and Chevalley-Shepard-Todd theorems, the Harish-Chandra isomorphism, Bernstein-Gelfand-Gelfand category O, projective functors, the nilpotent cone, the Duflo-Joseph and Duflo primitive ideal theorems, the Borel-Weil theorem, Springer resolution, and Beilinson-Bernstein localization via D-modules. It closes with the classification of representations of semisimple real groups through orbits on the flag variety. Materials follow MIT's OpenCourseWare model of lecture notes and problem sets, free to access under a Creative Commons license, with no certificate offered.