In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2, R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
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.
An irreducible finite-dimensional representation of a noncompact simple Lie group of dimension greater than 1 is never unitary.
(More precisely, the group SU(2) is simply connected and, although SL(2, R) is not, it has no non-trivial algebraic central extensions.)
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.
A major technique of constructing representations of a reductive Lie group is the method of parabolic induction.