.. _session_2B:
Online Training Session 2, Part B
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1. Summary of Session 2 Part A
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Welcome Back. Summary of previous hour.
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2. Minimal Principle Series of Split Groups
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Minimal Principal Series of split groups. :math:`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``
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3. More Functions
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More functions: ``cuspidal_data(Param)->(([int],KGBElt),Param)``.
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4. :math:`G=PSL(2,\mathbb{R})`
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G=PSL(2,R), parameters, composition series. Parameters of trivial rep; ``is_finite_dimensional(Param)->bool``, ``dimension(Param)->int``.
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5. :math:`G=Sp(4,\mathbb{R})`
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G=Sp(4,R). Principal series, tau-invariant: ``tau(Param)->[int]``; listing tau invariants, real roots types r1, r2, rn.
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6. Characters on Disconnected Part
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Question: Can lambda-rho get to the sign/trivial characters on the disconnected part?
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7. Lowest K Types
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Lowest K types: Principal series for Sp(4,R)
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8. :math:`G = SO(3,2)`
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G=SO(3,2): principal series, tau invariant, composition series
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9. More on :math:`Sp(4,\mathbb{R})`
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More on G=Sp(4,R). W-equivalent parameters. The function find([Param],Param)->int.
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10. :math:`E8`
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G=E8. Block sizes, ``split_form(InnerClass/RootDatum/Lietype)->RealForm``; ``real_forms(InnerClass/CartanClass)->[RealForm]``.
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