Representation Theory of SL_2

M390C. Spring Semester 2005. Tue-Th 11-12:30 RLM 9.166.

Office Hours:

W 3-4


The books by Carter et al. and Varadarajan are on 3-day loan reserve at the math physics library.

Brief description:

I will present an unorthodox introduction to the representation theory of Lie groups and Lie algebras, focussing entirely on the group of two by two matrices with determinant one. Thus we hope to cover a breadth of topics in representation theory, that are usually sacrificed for the depth of treating general Lie groups. We will start with finite dimensional representations of SL_2(C) (or equivalently SU_2) and their relation to the geometry of the Riemann sphere. We will then move on to unitary (infinite dimensional) representations of SL_2(C) and especially SL_2(R), and their relation to the geometry of the upper half plane and modular forms. We will (in an ideal universe) conclude with the beautiful geometry of the tree, associated to SL_2(Q_p), which is a p-adic analog of the upper half plane. Besides its intrinsic beauty, this subject is ubiquitous in number theory, topology and physics. In final projects we will explore some of these connections as well as generalizations to other Lie groups.


Basic Lie Theory:

Review Articles:

More advanced, SL_2 specific references: