liegr.. _session_3B:
Online Training Session 3, Part B
1. Questions
Welcome. Questions: Ordering of the KGB elements, closure relations of corresponding orbits, Hasse diagram for G=Sp(4,R).
2. More Slides
Slides, cont’d. More on discrete series parameters for Sp(4,R). Distinguished fiber. Borel subgroups containing the compact Cartan and corresponding discrete series.
3. Examples Using the Software
Examples using the software: SL(2,R), x=0->x=1, lambda->-lambda via W. Commands “print_KGB”, “block_of”, “show”, “hc_parameter”.
4. \(Sp(4,\mathbb{R})\)
Sp(4,R). How to identify each discrete series. Commands “all_discrete_series_gamma”, “K_0”, “status_texts”. Parameters associated to a KGB element and HC parameter: “discrete_series”; “generate_W”, “cross” (cross action of W on KGB elements. Stabilizer in W of a KGB element). Other choices of K<->KGB elements.
5. \(Sp(6,\mathbb{R})\)
G=Sp(6,R): Choice of x with prefered simple roots. “distinguished_fiber”, “rho_K”.
6. KGB on other Cartans
KGB on other Cartan subgroups. G= Sp(4,R). “print_real_Weyl”, KGB orbits of the real Weyl group.
7. Preview for Next Time
Preview for next time: General parameters: intermediate Cartans, cuspidal data, discrete series of levifactors.