 Representation Theory Of SL2(R)

# Representation theory of SL2(R)

to get instant updates about 'Representation Theory Of SL2(R)' on your MyPage. Meet other similar minded people. Its Free!

X

Description:
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:

[itex]H=begin0 & -i\ i & 0end[/itex] corresponding to the subgroup K of matrices [itex]begincos(theta) & -sin(theta)\ sin(theta)& cos(theta)end[/itex]
[itex]X=begin1 & i\ i & -1end[/itex]
[itex]Y=begin1 & -i\ -i & -1end[/itex]

The Casimir operator Ω is defined to be

[itex]Omega= H^2+1+2XY+2YX.[/itex]

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...

No feeds found

All Posting your question. Please wait!...