Representation theory of SL2(R)

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). == Structure of the complexified Lie algebra == We choose a basis H, X, Y for the complexification of the Lie algebra of SL(2, R) so that iH generates the Lie algebra of a compact Cartan subgroup K (so in particular unitary representations split as a sum of eigenspaces of H), and {H, X, Y} is an sl2-triple, which means that they satisfy the relations [ H , X ] = 2 X , [ H , Y ] = − 2 Y , [ X , Y ] = H .

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