Representation Theory of SL2(R) - Unitary Representations

Unitary Representations

The irreducible unitary representations can be found by checking which of the irreducible admissible representations admit an invariant positively-definite Hermitian form. This results in the following list of unitary representations of SL(2,R):

• The trivial representation (the only finite-dimensional representation in this list).
• The two limit of discrete series representations D₊₀, D₋₀.
• The discrete series representations D_k, indexed by non-zero integers k. They are all distinct.
• The two families of irreducible principal series representation, consisting of the spherical principal series I_+,iμ indexed by the real numbers μ, and the non-spherical unitary principal series I_-,iμ indexed by the non-zero real numbers μ. The representation with parameter μ is isomorphic to the one with parameter −μ, and there are no further isomorphisms between them.
• The complementary series representations I_+,μ for 0<|μ|<1. The representation with parameter μ is isomorphic to the one with parameter −μ, and there are no further isomorphisms between them.

Of these, the two limit of discrete series representations, the discrete series representations, and the two families of principal series representations are tempered, while the trivial and complementary series representations are not tempered.