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 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.