Representation Theory of SL2(R) - Structure of The Complexified Lie Algebra

Structure of The Complexified Lie Algebra

We choose a basis H, X, Y for the complexification of the Lie algebra of SL(2,R) so that iH generates the Lie algebra of a compact Cartan subgroup K (so in particular unitary representations split as a sum of eigenspaces of H), and {H,X,Y} is an sl₂-triple, which means that they satisfy the relations

One way of doing this is as follows:

corresponding to the subgroup K of matrices

The Casimir operator Ω is defined to be

It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL(2,R). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ2. Thus in the case of the Lie algebra sl₂, the infinitesimal character of an irreducible representation is specified by one complex number.

The center Z of the group SL(2,R) is a cyclic group {I,-I} of order 2, consisting of the identity matrix and its negative. On any irreducible representation, the center either acts trivially, or by the nontrivial character of Z, which represents the matrix -I by multiplication by -1 in the representation space. Correspondingly, one speaks of the trivial or nontrivial central character.

The central character and the infinitesimal character of an irreducible representation of any reductive Lie group are important invariants of the representation. In the case of irreducible admissible representations of SL(2,R), it turns out that, generically, there is exactly one representation, up to an isomorphism, with the specified central and infinitesimal characters. In the exceptional cases there are two or three representations with the prescribed parameters, all of which have been determined.