Cohomological Arthur Packets 2

Abstract: (This is a continuation of the talk from last week)

An important special case of Arthur packets are those of regular integral infinitesimal character. The trivial representation (attached to the dual principal nilpotent orbit) is an example.

It is known by a result of Salamanca that the unitary representations with regular integral infinitesimal character are precisely the cohomological representations. These are representations with non-trivial twisted $(\mathfrak g,K)$ cohomology. By a result of Vogan and Zuckerman these are precisely the modules $A_\mathfrak q(\lambda)$, constructed via cohomological induction from a unitary character of theta-stable Levi subgroup.

The conclusion is: assuming all is right with the world (i.e. Arthur's conjectures) an Arthur packet consisting of representations with regular integral infinitesimal character must consist of certain $A_\mathfrak q(\lambda)$-modules. These are sometimes referred to as "Adams-Johnson" packets; these were among the first interesting Arthur packets to be studied in the 1980s.

I'll discuss these things in the context of Atlas.

representation theory

Audience: advanced learners

( chat | slides | video )

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Real reductive groups/atlas

Series comments: Beginning January 6, 2022.

This is will be a working/learning seminar on (infinite-dimensional) representations of real reductive groups, aimed at grad students and researchers having some familiarity with representations of compact Lie groups. We'll use the atlas software; you should follow the directions at www.liegroups.org/ to install it on your laptop.

There is a space on slack

join.slack.com/t/atlasofliegro-tf77234/shared_invite/zt-10ic5x9hi-FxVZ1DFfUTDLEiWVHOBK2w

for questions and discussions about the seminar and the software.

The aim is for each seminar to last approximately one hour; the extra half hour in the schedule is meant to encourage lots of interaction with the audience. The idea of the seminar is that learning how the software does mathematical computations is an excellent way to understand the mathematics, as well as a great source of examples.

Notes for these seminars may be found in a OneNote notebook

1drv.ms/u/s!AuIZlbpNWacjghnk9A-T16rcHmBn

NOW (as of 12/1/22) notes moved to:

1drv.ms/u/s!AkgjPz9zobZTbXxynsk6bhvbINg

You should be able to access this link without any Microsoft account, and from it you can pass to the pages for each individual seminar. We will post with each individual seminar a direct link to the page for that seminar; but this direct link requires that you have a (free) Microsoft account. (Many of you will already have such an account, if you use ANY Microsoft software; and in that case the direct link will work without a problem.) We apologize for the inconvenience to the rest of you.

A good general introduction to what the seminar is about can be found at

www.liegroups.org/workshop2017/workshop/videos_and_computer

from a 2017 workshop. The mathematical subject matter is described in slides

www.liegroups.org/workshop2017/workshop/presentations/voganHO.pdf

from Vogan's lecture. The main ideas about how to realize this mathematics on a computer are described in Adams's lecture

www.liegroups.org/workshop2017/workshop/presentations/adams1HO.pdf

A quick introduction to the syntax for the software is in van Leeuwen's presentation

www.liegroups.org/workshop2017/workshop/presentations/vanLeeuwen.pdf

First goal is to learn how the software represents real reductive groups (precisely, the group of real points of any complex connected reductive algebraic group) and their representations; making sense of the software will lead to an understanding of the underlying mathematics. Second goal is to use the software to investigate experimentally questions about reductive groups.

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