Representation theory of SL2(R)

Representation Theory Of SL2(R)

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

<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>(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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