Representation Theory of SL2(R) - Principal Series Representations

Principal Series Representations

A major technique of constructing representations of a reductive Lie group is the method of parabolic induction. In the case of the group SL(2,R), there is up to conjugacy only one proper parabolic subgroup, the Borel subgroup of the upper-triangular matrices of determinant 1. The inducing parameter of an induced principal series representation is a (possibly non-unitrary) character of the multiplicative group of real numbers, which is specified by choosing ε = ± 1 and a complex number μ. The corresponding principal series representation is denoted I_ε,μ. It turns out that ε is the central character of the induced representation and the complex number μ may be identified with the infinitesimal character via the Harish-Chandra isomorphism.

The principal series representation I_ε,μ (or more precisely its Harish-Chandra module of K-finite elements) admits a basis consisting of elements w_j, where the index j runs through the even integers if ε=1 and the odd integers if ε=-1. The action of X, Y, and H is given by the formulas