Atlas of Lie Groups and Representations

Atlas of Lie Groups Software Package

Atlas of Lie Groups Software Package: Examples

Here are some examples to get you started with using the atlas software. There are also some extensive tables created by the software.

First of all some structure theory:

Simplest example: SL(2,R)
A non-simply connected group: PGL(2,R)
A reductive group: GL(2,R)
A complex group: SL(2,C)
A real semisimple group not inner to a compact group: SL(3)
The Weyl groups of the compact Cartans in SL(2,R) and PGL(2,R)
Interesting isogenies: Spin(3,1), SO(3,1) and PSO(3,1)
The real form of E₆ with maximal compact subgroup F₄

Now some representation theory:

The representations of SL(2,R) with infinitesimal character rho
The block of the trivial representation of PGL(2,R) (dual to the block of the trivial representation of SL(2,R))
The block of the trivial representation of Sp(4,R) and its dual block for SO(3,2)

Now some Kazhdan-Lusztig polynomials:

The block of the trivial representation of SL(2,R)
The block of the trivial representation of PGL(2,R) (dual to the previous one)
The block of the trivial representation of Sp(4,R) and its dual block for SO(3,2)
The block of the trivial representation of F₄ (split)

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