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Workshop on Unitary Representations of Real Reductive Groups

Talks on MTWF will take place in room L102 of the Warnock Engineering Building (WEB) . Talks on Thursday, July 4th, will take place in room 219 of the Leroy Cowles Building (LCB)

There will be three main lecture series (with more details provided below):

Jeffrey Adams: The Langlands classification
  A survey of the classification of irreducible representations, in a form suitable to computing Hermitian forms, and used by the atlas software. See the reading list for more details of necessary prerequisites.
Peter Trapa: Jantzen filtrations
  The main goal will be to explain the theory of the Janzten filtration for Harish-Chandra modules. This includes explaining the Jantzen conjecture (and its geometric interpretation) as well as understanding how signatures of the Jantzen form on layers of the Jantzen filtration behave under deformation. The main prerequistes are familiarity with the structure of complex semisimple Lie algebras, enveloping algebras, Category O, and Harish-Chandra modules for SL(2,R). See the reading list for more details.
David Vogan: Unitary representation of real reductive groups: computing Hermitian forms
  An introduction to unitary representations of real reductive Lie groups, aimed at understanding the algorithm for determining which representations are unitary. Will include some old material, like Knapp's description of unitary representations of non-real infinitesimal character, and some explanation of how characteristic p algebraic geometry gets into these analysis problems. The last lecture will describe some of the many interesting questions about which I know nothing.

In addition to sessions utilizing the atlas software, there will be several supplementary lectures, partly based on participants interests. Possible topics include the new "twisted" Lusztig-Vogan polynomials, Arthur packets and unipotent representations, etc. There will also be problems sessions in the evenings.
Because of the heat, we're considering scheduling the Thursday lectures at 3 and 4:30, so that people can go on an outing in the morning.

Monday Tuesday Wednesday Thursday Friday
9:30 Trapa 1Adams 2Adams 3 Adams 4Adams 5
11:00 Adams 1Trapa 3Vogan 3 Vogan 4Trapa 5
1:30 Trapa 2Vogan 2Trapa 4 Vogan 5
3:00 Vogan 1Supplement SupplementAdams 4?Supplement
4:30 SoftwareSoftwareSoftware Vogan 4?

Trapa: Jantzen Filtrations
1: Overview
2: The Janzten Filtration for Category O, Kazhdan-Lusztig Polynomials
3: The Janzten Filtration for Harish-Chandra modules, Kazhdan-Lusztig-Vogan Polynomials
4: Invariant Forms, Signature Characters, and Deformation
5: Application to Unitarity

Adams: The Langlands Classification
1: G-modules and (g,K)-modules
2: Discrete Series, tempered representations, induced representations
3: Structure theory, tori, irreducible and standard representations
4: Characters of tori and their covers, parameters
5: The Langlands classification

Vogan: Unitary representations
1: SL(2,R): how signatures change in the principal series
2: Abstract representations as complex variety, hermitian representations as real points. Knapp-Stein: most unitary representations are induced
3: Representations of real infinitesimal character and the extended group
4: Twisted KLV polynomials and the unitarity algorithm
5: Open problems: relations to orbit method; Jantzen forms in non-positive chambers; unitarity of unipotent representations...