Representation Theory of SL2(R) - Finite Dimensional Representations

Finite Dimensional Representations

For each nonnegative integer n, the group SL(2,R) has an irreducible representation of dimension n+1, which is unique up to an isomorphism. This representation can be constructed in the space of homogeneous polynomials of degree n in two variables. The case n=0 corresponds to the trivial representation. An irreducible finite dimensional representation of a noncompact simple Lie group of dimension greater than 1 is never unitary. Thus this construction produces only one unitary representation of SL(2,R), the trivial representation.

The finite-dimensional representation theory of the noncompact group SL(2,R) is equivalent to the representation theory of SU(2), its compact form, essentially because their Lie algebras have the same complexification and they are "algebraically simply connected". (More precisely the group SU(2) is simply connected and SL(2,R) is not, but has no non-trivial algebraic central extensions.) However, in the general infinite-dimensional case, there is no close correspondence between representations of a group and the representations of its Lie algebra. In fact, it follows from the Peter-Weyl theorem that all irreducible representations of the compact Lie group SU(2) are finite-dimensional and unitary. The situation with SL(2,R) is completely different: it possesses infinite-dimensional irreducible representations, some of which are unitary, and some are not.