Representation Theory
SL(2,R) is a real, non-compact simple Lie group, and is the split-real form of the complex Lie group SL(2,C). The Lie algebra of SL(2,R), denoted sl(2,R), is the algebra of all real, traceless 2 × 2 matrices. It is the Bianchi algebra of type VIII.
The finite-dimensional representation theory of SL(2,R) is equivalent to the representation theory of SU(2), which is the compact real form of SL(2,C). In particular, SL(2,R) has no nontrivial finite-dimensional unitary representations.
The infinite-dimensional representation theory of SL(2,R) is quite interesting. The group has several families of unitary representations, which were worked out in detail by Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
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