Atlas of Lie Groups and Representations

Notes from the AIM workshops.

Read me first

Real Forms and the Kac Classification (Jeffrey Adams)
Overview of the theory of real forms and strong real forms
Guide to the Atlas Software (from the 2007 Snowbird Conference) (Jeffrey Adams)
This is a good introduction to the math behind software. However it is extremely out of date, in that it uses the original interface designed by Fokko du Cloux, rather than the current atlas interface.
Unitary representations of real reductive groups (Jeffrey Adams, Peter Trapa, Marc van Leeuwen and David Vogan)
Complete and detailed description of the atlas algorithm for computing unitary representations. The Introduction is a good overview.
Infinite Dimensional Representations of Real Reductive Groups (David Vogan)
Notes from the Utah Workshop, 2012; overview of the preceding paper. Also see the slides from the Utah workshop
A Langlands Classification for Unitary Representations by David A. Vogan Jr., from Advanced Studies in Pure Mathematics, Volume 26 (1998), pp. 1-16.
Overview of work by Salamanca-Riba and Vogan on a conjectural description of the unitary dual.
Computing the Unitary Dual (David Vogan)
Overview of the atlas project from 2003; extremely out of date, especially in the computation of signatures of invariant forms.

Utah workshops

The atlas projected hosted two workshops for graduate students and postdocts, summer 2009 and summer 2013.

Annotated readling list from the 2009 workshop
Annotated readling list from the 2013 workshop
Notes from the 2009 workshop

Some more detail on the mathematics

Algorithms for Representation theory of Real Groups (Jeffrey Adams and Fokko du Cloux)
Fairly complete explanation of the basic atlas algorithm, up to but not including the KLV polynomials
Combinatorics for the Representation Theory of Real Groups (Fokko du Cloux)
Notes by Fokko du Cloux on the atlas algorithm. Somewhat out of date now (2005), but still useful for many fundamental algorithms for structure theory.
Equivalence of Parameters (Jeffrey Adams)
Detailed statement of the classification, including the precise notion of equivalence of parameters. This is particularly subtle (and critical) in the case of singular infinitesimal character.
Representations of K (David Vogan)
Detailed description of the irreducible representations of K in terms suitable for the Atlas. Also see notes by Peter Trapa from Palo Alto, 2005
Discrete Series and Characters of the Component Group (Jeffrey Adams)
Computing the signs which occur in endoscopic lifting of discrete series representations, in the context of the atlas algorithm. These are the "kappa" signs of Shelstad. Includes a self-contained description of the algorithm in the case of discrete series.
Computing Hodge Filtrations (Jeffrey Adams, Peter Trapa and David A. Vogan Jr.
Wilfried Schmid and Kari Vilonen have made a conjecture relating mixed Hodge modules and the unitary dual of a real reductive group. This article is intended to be progress in the direction of proving this conjecture. The main result is that the canonical Hodge filtration on a Harish Chandra module, when reduced modulo 2, gives the signature of the canonical c-form. This result would be a consequence of the main conjecture of Schmid and Vilonen.

Kazhdan-Lusztig-Vogan polynomials

Computing the Kazhdan-Lusztig algorithm (Fokko du Cloux)
Notes by Fokko du Cloux from 2005, revised 2011 to incorporate new recursion relations (see the next paper)
Improved Recursion Formulas for KLV Polynomials (David Vogan)
Improved recursion relations, which avoid the difficult "thickets" recursions of the original version.
Parameters for Twisted Representations (Jeffrey Adams and David Vogan)
Mathematical background behind the twisted Kazhdan-Lusztig-Vogan polynomials. This has appeared in the proceedings of the Vogan conference, and is also at arXiv:1502.03304
Computing Twisted KLV Polynomials (Jeffrey Adams)
Explicit recursion relations for the "twisted" KLV polynomials studied by Lusztig and Vogan. These are necessary for converting from c-invariant forms to ordinary Hermitian forms in the unequal rank case.
Implementation of the Kazhdan-Lusztig algorithm (Fokko du Cloux)
Technical notes about computing Kazhdan-Lusztig-Vogan polynomials for real groups. They were written by Fokko du Cloux for his own use.

Miscellaneous auxiliary papers

Computing Global Characters (Jeffrey Adams)
Using the atlas software to compute global characters.
The Contragredient (Jeffrey Adams and David Vogan)
The congtragredient (dual) representation corresponds to the Chevalley automorphism on the dual side. Includes a self-contained description of the Langlands classification over R.
Strong real forms and the Kac classification (Jeffrey Adams)
Expository treatment of the Kac classification of real forms.
Unitary Genuine Principal Series of the Metaplectic Group (Alessandra Pantano, Annegret Paul and Susana Salamanca-Riba)
Unitary minimal principal series of the metaplectic cover of Sp(2n,R).
The Omega-regular Unitary Dual of the Metaplectic Group (Allesandra Pantano, Annegret Paul, and Susana Salamanca-Riba)
Genuine unitary representations of the metaplectic group with (real) strongly regular infinitesimal character; analogous to Salamanca-Riba's classification of unitary representations of a linear group with (real) regular infinitesimal character.
Assigning Representation Parameters to Atlas Block Output (Annegret Paul)
Convert the output of the block command into something that is understandable by a human.
Disconnected Reductive Groups (David A. Vogan Jr.)
How to specify an arbitrary disconnected complex reductive group in a finite form.

Generalized Harish-Chandra modules

On Categories of Admissible (g,sl(2)) modules (Ivan Penkov, Vera Serganova and Gregg Zuckerman)
Equivalence of categories between a certain parabolic category of g-modules and admissible (g,sl(2))-modules.
Generalized Harish-Chandra modules (Gregg Zuckerman)
Introduction to generalized Harish-Chandra modules, which are a generalization of the representations studied by Atlas in the context of the unitary dual. Includes an introduction to the Zuckerman functor.
Algebraic Methods in the Theory of Harish-Chandra modules (Ivan Penkov and Gregg Zuckerman)
Overview of the theory of generalized Harish-Chandra modules.
On the Structure of the Fundamental Series of Generalized Harish-Chandra Modules (Ivan Penkov and Gregg Zuckerman)
Fundamental series for generalized Harish-Chandra modules.

Technical details

These papers of interest mostly to experts.
Hermitian forms for Sp(4,R) (Jeffrey Adams and Annegret Paul)
Extensive example of the computation of Hermitian forms for Sp(4,R), in the language of atlas.
Branching Example: Sp(4,R) (David Vogan)
An example of how the software computes representations of K.
Some Notes on Parametrizing Representations (Jeffrey Adams)
For the experts; some technical issues involving parametrizing representations and the "basepoint" issue.
Notes on the Hermitian Dual (Jeffrey Adams)
Computing the Hermitian dual involution in the setting of atlas.
Notes on Doubly Extended Groups (Jeffrey Adams)
Incorporating the Hermitian dual in the doubly extended framework.
Computing y (Jeffrey Adams)
If you have to ask...