Computing unitary duals, I: cohomological induction

Abstract: We are planning to conclude this seminar (for now?) with three talks January 12, 19, and 26, 2023. Topic is progress and plans for the original goal of the atlas project: to make software that can describe the unitary dual of any real reductive group G(R).

There are two fundamental classical techniques to construct unitary representations. The first (due to Israel Gelfand and his collaborators) is real parabolic induction. A theorem in Knapp's "Overview" book gives a very simple way to identify most of the unitary representations that can be obtained in this way: they are the ones with non-real infinitesimal character. In light of that theorem, one can study _only_ representations with REAL infinitesimal character.

The second classical technique (due to Gregg Zuckerman and those who stole from him) is cohomological parabolic induction. The analogue of the theorem in Knapp's book would say that any unitary representation with non-imaginary infinitesimal character can be obtained by cohomological induction. THIS IS NOT TRUE, but it is nearly true.

What's actually true is that any unitary representation for which the real part of the infinitesimal character is LARGE ENOUGH can be obtained by cohomological induction. The question of what "large enough" means is best expressed in terms of "nonunitarity certificates. Today I will state these results with some care, and start to look at nonunitarity certificates.

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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