Abstract: Let SL(2,R) be the set of all two-by-two matrices over the real numbers with determinant 1. This expository article describes a representation theory for the universal covering group G of SL(2,R) which is motivated by a localization theory due to A. Beilinson and J. Bernstein. We explicitly describe the irreducible representations of G using objects familiar to the average calculus student -- the complex plane, vector spaces, the algebra of two-by-two matrices, the derivative of a rational polynomial function -- and some elementary facts about Lie group representations and algebraic D-modules. In particular, we will realize the irreducible representations of G as global sections of certain D-modules on the flag variety of sl(2,C) -- the Lie algebra of complex two-by-two matrices with trace zero.
There is also a Java applet which allows you to explore the reducibility of the standard modules and the geometry of the irreducible representations.