Unitary Representation - Unitarizability and The Unitary Dual Question

Unitarizability and The Unitary Dual Question

In general, for non-compact groups, it is a more serious question which representations are unitarizable. One of the important unsolved problems in mathematics is the description of the unitary dual, the effective classification of irreducible unitary representations of all real reductive Lie groups. All irreducible unitary representations are admissible (or rather their Harish-Chandra modules are), and the admissible representations are given by the Langlands classification, and it is easy to tell which of them have a non-trivial invariant sesquilinear form. The problem is that it is in general hard to tell when the quadratic form is positive definite. For many reductive Lie groups this has been solved; see representation theory of SL2(R) and representation theory of the Lorentz group for examples.