Papers and notes

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 the 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 However it is extremely out of date, in that it uses the original interface designed by Fokko du Cloux, rather than the current atlas interface.

arXiv:0807.3095

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.

arXiv:1212.2192

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

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Overview of the atlas project from 2003; extremely out of date, especially in the computation of signatures of invariant forms.

The FPP Conjecture for Real Reductive Groups

Dougal Davis and Lucas Mason-Brown

Proof of the FPP conjecture, giving a strong upper bound on the unitary dual of a real reductive group.

arXiv:2411.01372

Hodge Theory, Intertwining Functors, and the Orbit Method for Real Reductive Groups

Dougal Davis and Lucas Mason-Brown

The Hodge filtrations of Schmid and Vilonen on unipotent representations of real reductive groups.

arXiv:2503.14794

Unitary Representations of Real Groups and Localization Theory for Hodge Modules

Dougal Davis and Kari Vilonen

Proof of the conjecture of Schmid and Vilonen that the unitarity of a representation with real infinitesimal character can be read off from its Hodge filtration.

arXiv:2309.13215

The Unitarity of Arthur Packets for Real Reductive Groups

Jeffrey Adams, Andrei Ionov, Lucas Mason-Brown and David Vogan

Proof of the longstanding conjecture that all Arthur packets for real reductive groups consist of unitary representations.

arXiv:2606.01609

Lowest K-types in the Local Langlands Correspondence

Jeffrey Adams and Alexandre Afgoustidis

A natural connection between the lowest K-types of a representation of a real reductive group and its Langlands parameters.

arXiv:2402.03552

Nilpotent Invariants for Generic Discrete Series of Real Groups

Jeffrey Adams and Alexandre Afgoustidis

Representations of real groups which admit Whittaker models.

arXiv:2410.04134

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.

arXiv:0807.3093

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.

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.

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.

arXiv:1710.04709

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.

Computing Global Characters

Jeffrey Adams

Using the atlas software to compute global characters.

The Contragredient

Jeffrey Adams and David Vogan

The contragredient (dual) representation corresponds to the Chevalley automorphism on the dual side. Includes a self-contained description of the Langlands classification over R.

arXiv:1201.0496

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

Alessandra 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.

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.

arXiv:1604.04672

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.

arXiv:1310.8058

On the Structure of the Fundamental Series of Generalized Harish-Chandra Modules

Ivan Penkov and Gregg Zuckerman

Fundamental series for generalized Harish-Chandra modules.

arXiv:1109.1804

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…

Notes from the AIM workshops, 2003–2008

By workshop: Palo Alto, July 2008 · Maryland, March 2008 · Palo Alto, July 2007 · MIT, March 2007 · Palo Alto, July 2006 · Palo Alto, July 2005 · Palo Alto, July 2004 · Palo Alto, Summer 2003

Utah workshops, 2009 and 2013

The atlas project hosted two workshops for graduate students and postdocs, summer 2009 and summer 2013.

Annotated reading list from the 2009 workshop · Annotated reading list from the 2013 workshop · Notes from the 2009 workshop