 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).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

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