MAT 1197S - Representations of reductive p-adic groups

Instructor: Fiona Murnaghan

Class times:

Monday 2:30-4:30; Wednesday 3:10-4, 215 Huron

March 5: Section 8(depth zero supercuspidals) is complete.

April 23: Section 9 has been added, and part of Section 10.

Course notes.

Problems on representations of the mirabolic subgroup.

Material covered to date:

Basic results about smooth representations (section 3)
Haar measure; convolution of smooth locally constant functions;
admissible representations; characters (section 4);
general properties of induced representations (section 5).
Section 6 - parabolic subgroups, Iwahori decomposition, related integral formulas, Jacquet modules, admissibility of Jacquet modules.
Basic lemmas about supercuspidal representations.
A smooth representation is supercuspidal if and only if all of its Jacquet modules are zero.
Transitivity of Jacquet modules.
Jacquet's subrepresentation theorem.
An irreducible smooth representation of a reductive p-adic group is admissible (proof for general linear group).
Induced representations that are supercuspidal.
Parahoric subgroups of general linear groups;
Depth zero representations.
Depth zero supercuspidal representations.
Definition and basic properties of cuspidal representations of reductive groups over finite fields.
Depth zero unrefined minimal K-types.
General form of depth zero supercuspidals.
Parahoric subgroups of symplectic groups.
Construction of Deligne-Lusztig - representations of reductive groups over finite fields that are associated to characters of tori.
Parametrizaton of conjugacy classes of maximal tori in reductive groups over finite fields.
Elliptic maximal tori in symplectic groups over finite fields.
Examples of elliptic maximal tori in symplectic p-adic groups.
Definition of Weil group of a p-adic field F, inertia group, etc.
Comments about local Langlands conjectures.
Brief comments about L-groups.
Tame regular semisimple Langlands parameters.
Example: L-packet of depth zero supercuspidal representations of 4 by 4 symplectic group.
Depth zero L-packets constructed by DeBacker and Reeder; characters and stability.
Local Langlands conjectures and supercuspidal representations.