Representation Theory Of SL2(R)

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In mathematics, the main results concerning irreducible **unitary representations** of the Lie group SL<sub>2</sub> 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<sub>2</sub>(**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 is an sl<sub>2</sub>-triple, which means that they satisfy the relations

One way of doing this is as follows:

The Casimir operator Ω is defined to be

It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL<sub>2</sub>(**R**). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ<sup>2</sup>. Thus in the case of the Lie algebra sl<sub>2</sub>, the infinitesimal character of an irreducible representation is specified...

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- <math> =2X, quad =-2Y, quad =H. </math>

One way of doing this is as follows:

- <math>H=begin0 & -i\ i & 0end</math> corresponding to the subgroup
*K*of matrices <math>begincos(theta) & -sin(theta)\ sin(theta)& cos(theta)end</math> - <math>X=begin1 & i\ i & -1end</math>
- <math>Y=begin1 & -i\ -i & -1end</math>

The Casimir operator Ω is defined to be

- <math>Omega= H^2+1+2XY+2YX.</math>

It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL<sub>2</sub>(

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