Papers & Notes
Research Papers
Papers, workshop notes, and expository material produced by the Atlas project. See also notes from the AIM workshops.
Introductory & Overview
Real Forms and the Kac Classification
Overview of the theory of real forms and strong real forms.
Guide to the Atlas Software Outdated interface
Good introduction to the math behind the software, but uses the original Fokko du Cloux interface rather than the current atlas interface.
Unitary Representations of Real Reductive Groups
Complete and detailed description of the Atlas algorithm for computing unitary representations. The Introduction provides a good overview.
Infinite Dimensional Representations of Real Reductive Groups
Notes from the Utah Workshop; overview of the preceding paper. Also see the slides from the Utah workshop.
A Langlands Classification for Unitary Representations
Overview of work by Salamanca-Riba and Vogan on a conjectural description of the unitary dual.
Computing the Unitary Dual Outdated
Overview of the Atlas project from 2003; now out of date, especially regarding computation of signatures of invariant forms.
Algorithms & Structure Theory
Algorithms for Representation Theory of Real Groups
Fairly complete explanation of the basic Atlas algorithm, up to but not including the KLV polynomials.
Combinatorics for the Representation Theory of Real Groups
Notes on the Atlas algorithm. Somewhat out of date but still useful for fundamental algorithms for structure theory.
Equivalence of Parameters
Detailed statement of the classification, including the precise notion of equivalence of parameters — particularly subtle in the case of singular infinitesimal character.
Representations of K
Detailed description of the irreducible representations of K in terms suitable for the Atlas.
Discrete Series and Characters of the Component Group
Computing the signs which occur in endoscopic lifting of discrete series representations in the Atlas algorithm context.
Computing Hodge Filtrations
Progress toward proving the Schmid–Vilonen conjecture relating mixed Hodge modules and the unitary dual of a real reductive group.
Kazhdan–Lusztig–Vogan Polynomials
Computing the Kazhdan–Lusztig Algorithm
Revised to incorporate new recursion relations (see next paper).
Improved Recursion Formulas for KLV Polynomials
Improved recursion relations which avoid the difficult "thickets" recursions of the original version.
Parameters for Twisted Representations
Mathematical background behind the twisted KLV polynomials. Also at arXiv:1502.03304.
Computing Twisted KLV Polynomials
Explicit recursion relations for the "twisted" KLV polynomials, necessary for the unequal rank case.
Implementation of the Kazhdan–Lusztig Algorithm
Technical notes on computing KLV polynomials for real groups, written by Fokko du Cloux for his own use.
Miscellaneous
Computing Global Characters
Using the Atlas software to compute global characters.
The Contragredient
The contragredient (dual) representation corresponds to the Chevalley automorphism on the dual side. Includes a self-contained description of the Langlands classification over ℝ.
Assigning Representation Parameters to Atlas Block Output
How to convert the output of the block command into human-readable form.
Disconnected Reductive Groups
How to specify an arbitrary disconnected complex reductive group in finite form.
Utah Workshops — Graduate Reading Material
Annotated Reading List — 2009 Workshop
Reading list for graduate students and postdocs from the 2009 summer workshop.
Notes from the 2009 Workshop
Schedule and notes from the 2009 summer workshop lectures.
Annotated Reading List — 2013 Workshop
Reading list from the 2013 follow-up workshop.