# 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].