Online Training Session 2, Part B

1. Summary of Session 2 Part A

Welcome Back. Summary of previous hour.

2. Minimal Principle Series of Split Groups

Minimal Principal Series of split groups. \(SL(2,\mathbb{R})\) example. Use of functions KGB(RealForm,int)->KGBElt, involution(KGBElt)->mat, all_parameters_gamma(RealForm,ratvec->[Param]), Induced representations: I(Param)->(Param,string), composition_series(Param)->ParamPol, show(ParamPol)->void

3. More Functions

More functions: cuspidal_data(Param)->(([int],KGBElt),Param).

4. \(G=PSL(2,\mathbb{R})\)

G=PSL(2,R), parameters, composition series. Parameters of trivial rep; is_finite_dimensional(Param)->bool, dimension(Param)->int.

5. \(G=Sp(4,\mathbb{R})\)

G=Sp(4,R). Principal series, tau-invariant: tau(Param)->[int]; listing tau invariants, real roots types r1, r2, rn.

6. Characters on Disconnected Part

Question: Can lambda-rho get to the sign/trivial characters on the disconnected part?

7. Lowest K Types

Lowest K types: Principal series for Sp(4,R)

8. \(G = SO(3,2)\)

G=SO(3,2): principal series, tau invariant, composition series

9. More on \(Sp(4,\mathbb{R})\)

More on G=Sp(4,R). W-equivalent parameters. The function find([Param],Param)->int.

10. \(E8\)

G=E8. Block sizes, split_form(InnerClass/RootDatum/Lietype)->RealForm; real_forms(InnerClass/CartanClass)->[RealForm].