Introduction to Lie groups II: Representation theory og compact Lie groups

Lecturer: Paul Friedman

Content:
Group representations are homomorphisms of a group into bounded linear operators on a complex Hilbert space, usually with some continuity property. The applications of representation theory stretch across many fields of mathematics, including harmonic analysis, number theory, differential geometry and mathematical physics.

The building blocks of the theory are irreducible representations. For compact connected Lie groups, the irreducible representations are fully understood. The centerpieces of this theory are the Theorem of the Highest Weight and the Weyl Character Formula.

The topics we will cover in the course are likely to include: differential forms, Haar measure for Lie groups, representations of compact groups, including the Peter-Weyl Theorem, the Theorem of the Highest Weight, the Weyl Character Formula and the Borel-Weil Theorem. Time permitting, we will develop some structure theory of noncompact Lie groups.

Prerequisites: Introduction to Lie groups(Fall 2000). Newcomers are welcome, but should speak with the instructor first.

Literature: A. Knapp, Lie Groups Beyond an Introduction, Birkh user, 1996. Notes.