Classifying the unitary dual (part 1 of infinitely many...)
David Vogan (MIT)
Abstract: We've explained a lot about how atlas can offer information about individual representations, and certainly how it can check whether an individual representation is unitary. Want to start talking about how to describe the full unitary dual: to give a finite and complete description of the answers to infinitely many questions "is it unitary?" The Dirac inequality from last week is useful hint; I will try to say how it can be part of a general picture of the unitary dual, and how it might usefully be modified.
representation theory
Audience: advanced learners
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.
Organizers: | Jeffrey Adams*, David Vogan* |
*contact for this listing |