Online Training Session 4, Part A

1. Welcome Back

Welcome. Outline. Parameters p=(x,lambda, nu). Infinitesimal character. Equivalence of parameters. Canonical bijections parameters<->std, irr modules. Composition series, Character formula, KL polynomials.

2. New atlas Version

New atlas version 0.6.3 and new .at files.

3. Review of \(SL(2,\mathbb{R})\)

Example: Review of G=SL(2,R), trivial; DS, Harish Chandra parameters; block of the trivial.

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

Example: PGL(2,R): trivial; DS, effect of disconnectedness.

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

Example: Sp(4,R). block_of_the_trivial(G). Representations of G associated to intermediate Cartans. Example:element 4 in block, KGBElt x=5: cuspidal_data command: Parabolic subgroup P and (relative) DS representation of Levi factor M attached to an atlas parameter p. Levi command.

6. \(Sp(4,\mathbb{R})\), KGBElt x=5

Example Sp(4,R), KGBElt x=5 continued: induce_standard vs induce_irreducible. show(composition_series( )) and show(character_formula(( ))

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

Sp(4,R), continued. Cross action, representations on same Cartan and same Levi factor: elements 4,5,7,8 in the block, KGBElts x=5,6,7,8.

8. Representations of \(Sp(4,\mathbb{R})\)

Sp(4,R): Reps. on the other intermediate Cartan: elements 6,9; KGBElts x=4,9. Summary of all reps in block.