# Real reductive groups/atlas

representation theory

 Audience: Advanced learners Seminar series time: Thursday 14:30-16:00 in your time zone, UTC Organizers: Jeffrey Adams*, David Vogan* *contact for this listing

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

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

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.

Upcoming talks
Past talks
ThuSep 2914:30David VoganDuality, associated varieties, and nilpotent orbits
ThuSep 2214:30Jeffrey AdamsCohomological Arthur packets 3/Open Mic NightI'l
ThuSep 1514:30Jeffrey AdamsCohomological Arthur Packets 2
ThuSep 0114:30David VoganMore about discrete series restriction
ThuAug 2514:30David VoganCohomological induction and restricting discrete series to K
ThuAug 1814:30Jeffrey AdamsHermitian forms on finite- dimensional representations
ThuAug 1114:30David VoganDuality for singular integral infinitesimal character
ThuAug 0414:30David VoganDuality for singular and non-integral infinitesimal character
ThuJul 1414:30NONETAKE A WEEK OFF
ThuJul 0714:30David VoganAffine Weyl group facets and the unitary dual
ThuJun 3014:30David VoganClassifying the unitary dual (part 1 of infinitely many...)
ThuJun 2314:30David VoganDirac operator in atlas
ThuJun 1614:30ONE WEEK BREAKNo seminar this Thursday, resuming next week
ThuJun 0914:30Jeffrey AdamsJantzen filtration and open mic night
ThuJun 0214:30Jeffrey AdamsMore loose ends: translation, Jantzen filtration
ThuMay 2614:30Jeffrey AdamsLoose ends: Hermitian representations, more on parameters, translation and the Jantzen filtration
ThuMay 1914:30David VoganUnitary dual of F4_B4 in atlas
ThuMay 1214:30David VoganUnitary dual of SO(2n,1) in atlas
ThuMay 0514:30Jeffrey AdamsTheta-stable parabolic subgroups and cohomological induction in atlas
ThuApr 2814:30David VoganReal parabolic subgroups and induction in atlas
ThuApr 2114:30Jeffrey AdamsLusztig's parametrization of families
ThuApr 0714:30David VoganNilpotent orbits and atlas
ThuMar 3114:30David VoganGelfand-Kirillov dimension and atlas
ThuMar 2414:30Jeffrey AdamsWeyl group representations and atlas II
ThuMar 1714:30Jeffrey AdamsWeyl group representations and atlas
ThuMar 1015:30David VoganHow atlas does what it says its doing
ThuMar 0315:30Jeffrey AdamsSignature character formulas and unitary representations 2
ThuFeb 2415:30Jeffrey AdamsSignature character formulas and unitary representations (or why we needed this software in the first place)
ThuFeb 1715:30David VoganCharacter formulas and Kazhdan-Lusztig polynomials (or why we needed this software in the first place)
ThuFeb 1015:30Jeffrey AdamsThe Atlas Way (More on KGB)
ThuFeb 0315:30David VoganUnderstanding $K$: how atlas understands Cartan's theory of maximal compact subgroups
ThuJan 2715:30Jeffrey AdamsBranching to $K$: writing an infinite-dimensional representation of $G$ as a sum of finite-dimensionals of $K$
ThuJan 2015:30David VoganParameters: how atlas understands a representation
ThuJan 1315:30Jeffrey AdamsRoot data: how atlas understands a reductive group
ThuJan 0615:30David VoganWhat groups? What representations? What software?
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