Online Training Session 2, Part A¶

1. New Features on Website¶

Welcome back. New features in Atlas website. Other videos.

2. Questions from Session 1¶

Questions from last time. Coordinates in atlas, change of coordinates matrix, the coordinates.at file. Newest version of atlas-scripts.

3. Cartans¶

Cartans of different real forms. G= SO(5,4). occurrence_matrix

4. Parameters¶

Parameters.Parameter space, irreducible and standard modules. Langlands parameters.

5. Example: $$SL(2,\mathbb{R})$$¶

Example: SL(2,R). The command “all_parameters_gamma”

6. Characters of Tori¶

Characters of real tori. The command “torus”. Components of the characters: {theta, gamma, lambda} and {theta,lambda,nu}. Example: Characters of $$T=S^1$$. Equivalent parameters.

7. $$T = \mathbb{R}^x$$¶

Example: Characters of T=R^x.

8. $$T = \mathbb{C}^x$$¶

Example: Characters of T=C^x

9. Differential¶

Differential in terms of lambda and nu. The functions x(parameter), lambda(parameter), nu(parameter), involution(x)

10. Trivial Representation of $$SL(2,\mathbb{R})$$¶

Example: trivial representation of SL(2,R)

11. Preview: KGB Orbits¶

Preview: x –> K orbit on G/B.