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.