Cohomological induction and restricting discrete series to K

Abstract: Harish-Chandra's theorem (and also the Langlands classification, which is based on the theorem) says that a discrete series representation of a real reductive G is specified by a character of a compact Cartan subgroup T of G. This talk is about Zuckerman's idea of how to implement that: given a character of a compact Cartan, how to construct a (g.K) module. The construction is most explicit as a representation of K; so I'll talk about how to see the restriction to K of a discrete series, and what things we do and don't know about that.

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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