Papers by David Vogan (...and his friends)

• Langlands philosophy and representation theory
  

  Delivered at Academia Sinica, 19 June 2024; and at Xiamen University, 19 July 2024.
  Quick summary of what the Langlands program has to say about representations of real reductive groups.
  
  Version of July 19, 2024
  xiamen24
  

• What does the unitary dual look like?
  

  Delivered at Conference on Real Reductive Groups and Theta Correspondencee, Tianyuan Mathematics Research Center, 22 July, 2024.
  Formulation of the FPP Conjecture about unitary representations.
  At the end of these slides, I did a little demo of the atlas software. The second file below is a screenshot of that.
  
  Version of July 22, 2024
  FPP24HO
  atlas demo
  

• Generalizing endoscopy
  

  Delivered in person at the 18th Representation Theory Conference, Dubrovnik, 19 June, 2023.
  Delivered in person at the Harish-Chandra Centenary Conference, Harish-Chandra Research Institute, Prayagraj, India, October 13, 2023.
  Explains a way to significantly enlarge the number of endoscopic groups, while still preserving some of their attractive representation-theoretic properties.
  Talk is entirely about p-adic groups, although the only theorems (joint with Jeffrey Adams and Lucas Mason-Brown) concern real groups.
  
  Version of June 14, 2023
  dubrovnik23HO
  
  Version of October 12, 2023
  HC10.23HO
  

• How to compute the unitary dual
  

  Delivered by zoom at Zhejiang University, 23 February, 2023
  
  Microsoft OneNote notes written during talk.
  
  Version of February 22, 2023
  zhejiang23HO
  

• Nonunitarity certificates
  

  Delivered at William and Mary, conference "Noncommutative geometry and harmonic analysis on homogeneous spaces," January 16-20 2023.
  
  A better title would have been "introduction to the Langlands classification for real reductive groups, because there is virtually nothing about nonunitarity certificates in the slides.
  
  Version of January 16, 2023
  NCGAHS_HO
  

• Enumeration and unitarity of Arthur's representations for exceptional real groups
  

  Delivered at JMM 2023, special session "Automorphic forms and representation theory."
  
  In addition to the pdf file of slides, there are two pages of Microsoft OneNote scribbles with an outline of examples of the results for PGL(2). These slides should be viewable in a web browser.
  
  Once you have opened the OneNote page in a browser, you will see on the left links to other pages. These clog up the screen; you can hide them by clicking the "shelf of books" icon in the upper left.
  
  I have also tried to insert this link in the pdf file, on pages 14 and 18.
  
  Version of January 7, 2023
  JMM23HOB.pdf
  

• Affine Weyl group alcoves
  

  Colloquium delivered at UMassBoston October 12, 2022.
  
  Version of October 12, 2022
  affineWHO.pdf
  

• How to compute unitary representations
  

  Lecture delivered in person at 17th Representation Theory conference, Dubrovnik, October 3-7, 2022.
  
  Version of October 4, 2022
  computeUnitary.pdf
  

• The unitary dual problem
  

  Lecture delivered in person at 7th Croatian Mathematical Congress, Split, June 15-18
  
  Version of June 17, 2022
  splitDVHO.pdf
  

• Characters of unipotent representations
  

  Lecture delivered in person at "Symmetry in Geometry and Analysis," Reims, June 7-11, for the 60th birthday of Toshiyuki Kobayashi.
  
  Version of June 6, 2022
  kobayashiDVHO.pdf
  

• What's special about special?
  

  Lecture by zoom to special session of AMS sectional meeting. Topic is special nilpotent orbits, and an "integrality" definition of special proposed by Meinolf Geck and proven by Geck, Junbin Dong, and Gao Yang. The purpose of the talk is to advertise the problem of proving directly that the associated variety of a primitive ideal of integral infinitesimal character must satisfy Geck's integrality condition. To be delivered March 19, 2022.
  
  Version of March 16, 2022
  specialAMS3.22HO.pdf
  

• Unipotent representations of complex reductive groups
  

  Lecture by zoom to the Automorphic Project (Johns Hopkins/University of Michigan) asking forgiveness for a 1985 paper with Dan Barbasch. The part actually about that paper is unfinished, which is perhaps OK since the paper is finished. Delivered December 10, 2021.
  
  Version of December 10, 2021; small typos corrected.
  barbaschVoganHO.pdf
  Microsoft OneNote page with a few words about relating W reps to nilpotent orbits relevant to talk.

• Structure of Harish-Chandra cells
  

  Lecture (in person!) to the UMass Amherst representation theory seminar, introducing the notion of "Harish-Chandra cells" (a generalization of Kazhdan-Lusztig left cells). Delivered September 13, 2021.
  
  Version of September 12, 2021
  HCwcellsUMASSHO.pdf
  

• Representations of reductive groups
  

  Two lectures to the AIM research group RTNCG, meant to introduce real reductive group representations to C^*-algebraists. Delivered via Zoom at August 16, 18, 2021.
  
  Version of August 18, 2021
  voganSchoolHO.pdf
  

• Understanding reductive group representations
  

  The Langlands classification provides a fairly easy and explicit list of irreducible representations of a real reductive group G. This talk concerns the question of how to find detailed information about a representation from its Langlands parameter. Delivered via Zoom at Nankai University July 23, 2021.
  
  Version of July 17, 2021
  nankai21HO.pdf
  

• Branching representations to K
  

  AIM's working group "Representations and noncommutative geometry," see this link, has a weekly half hour Zoom presentation called "What I do." Each presenter is supposed to expound on this topic, with a view toward elucidating common interests. I decided that what I do is restrict real reductive group representations to maximal compact subgroups. The slides were borrowed from slides that I've used on (many) previous occasions, and I left many old slides in the pack with no intention of presenting them. Delivered (via Zoom) March 15, 2021.
  
  Version of March 14, 2021
  voganRTNCOGHO.pdf
  

• Weyl group representations and Harish-Chandra cells
  

  Review of Joseph's work on primitive ideals, associated varieties, and left cells for highest weight modules; Lusztig's detailed description of left cells; and some hints of extensions to Harish-Chandra modules. Delivered (via Zoom) at the Weizmann Institute March 10, 2021.
  
  Version of March 10, 2021
  weizmannHO.pdf
  

• Lie groups and representations
  

  A very quick and broad introduction to what Lie groups and representation theory are about, and what kinds of questions they can answer. Delivered (via Zoom) at Soochow University October 20, 2020.
  
  Version of October 12, 2020
  soochowHO.pdf
  

• Unitary representations and bottom layer K-types
  

  From a lecture introducing the "bottom layer method" for classifying unitary representations. The slides are very incomplete; hope to add to them later. Delivered at "Rutgers Mini-workshop on the Unitary Dual Problem" January 29, 2020; organized by Steve Miller.
  
  Version of January 31, 2020
  bottomHO.pdf
  

• Local Langlands conjecture for finite groups of Lie type
  

  Ideas for a Langlands classification of representations of a finite Chevalley group, and connections with p-adic local Langlands.
  
  Version of January 17, 2020
  finiteLanglandsAMS20HO.pdf
  

• Dirac index and associated cycles for Harish-Chandra modules
  Joint with Salah Mehdi, Pavle Pandzic, and Roger Zierau
  

  Proves the relationship between the index of the Dirac operator (for real reductive group representations) and multiplicities of certain orbits in the associated cycle of the representation, conjectured in the paper with Mehdi and Pandzic below.
  
  To appear in Advances in Mathematics
  
  Version of October 11, 2019, 30 pages.
  mpvzREV3.pdf
  

• Associated varieties for real reductive Lie groups
  Joint with Jeffrey Adams
  

  We give an algorithm to calculate the associated variety of any simple Harish-Chandra module for a linear reductive group. The algorithm has been implemented by Adams in the atlas software.
  
  To appear in Pure and Applied Mathematics Quarterly.
  
  Version of February 9, 2019, 76 pages.
  AdamsVoganPAMQ.pdf
  Version of March 29, 2021, 77 pages; lots of typos corrected, with thanks to careful referees.
  AdamsVoganPAMQarx2.pdf
  

• Signatures for finite-dimensional representations of real reductive Lie groups
  Joint with Daniil Kalinov and Christopher Xu
  

  Many finite-dimensional representations of a real reductive Lie group carry invariant Hermitian forms. We give a simple formula for the signature, analogous to the Weyl dimension formula.
  
  Version of September 10, 2018, 36 pages.
  finitesig.pdf
  

• Signatures for finite-dimensional representations
  

  Slides for a talk based on the paper above; given for Lie Theory Day in mathematics department at National University of Singapore, March 22, 2019.
  
  Version of March 22, 2019.
  finisigHO.pdf
  

• Quantization, the orbit method, and unitary representations
  

  Slides for a talk at Representation theory, geometry, and quantization: the mathematical legacy of Bertram Kostant, May 28-June 1, 2018.
  
  Version of May 28, 2018.
  kostant18HO.pdf
  

• Laplacians on spheres
  Joint with Henrik Schlichtkrull and Peter Trapa
  

  For each way of writing a sphere as a homogeneous space G/H for compact groups, recalls the G-decomposition of L^2(G/H); then examines consequences for other real forms of G/H.
  
  Sao Paulo J. Math. Sci. 12 (2018), no. 2, 295-358
  
  Version of March 3, 2018, 49 pages.
  spheres.pdf
  

• The size of infinite-dimensional representations
  

  Based on Takagi Lectures below.
  
  Jpn. J. Math. 12 (2017), no. 2, 175-210.
  
  Version of August 2, 2017 (small corrections), 35 pages.
  takagiB.pdf
  

• The size of infinite-dimensional representations
  

  Slides for two talks at the 18th Takagi Lectures at the University of Tokyo November 5-6, 2016. Subject was defining and computing Gelfand-Kirillov dimension for representations, especially of reductive Lie groups.
  
  Version of November 7, 2016.
  sizeI.pdf
  sizeII.pdf
  Version delivered at CIRM conference "Geometric quantization and applications." October 12, 2018.
  sizeCIRMHO.pdf
  

• Translation principle for Dirac index
  Joint with Salah Mehdi and Pavle Pandzic
  

  Formulates a conjectural relationship between the index of the Dirac operator (for real reductive group representations) and multiplicities of certain orbits in the associated cycle of the representation.
  
  Amer. J. Math. 139 (2017), no. 6, 1465-1491.
  
  Version of April 2, 2016, 23 pages.
  index_translation_2April2016.pdf
  

• Langlands parameters and finite-dimensional representations
  

  Slides for talk at the workshop "New Developments in Representation Theory," IMS, Singapore. Subject is the use of Langlands' ideas to describe finite-dimensional representations of compact groups: maximal compact subgroups of groups over local fields, and finite groups of Lie type.
  
  Version of March 20, 2016.
  finiteLanglandsHO.pdf
  Version of April 7,2019 TORA X conference, correcting some moderately serious errors in the formulations of the main conjectures.
  finiteLanglandsTORAHO.pdf
  

• Conjugacy classes and group representations
  

  Slides for talk at the Joint Mathematics Meetings in Seattle, January 7, 2016. Subject is the relationship between the two things in the title, with emphasis on the Kirillov-Kostant "method of coadjoint orbits" (but with discussion of the case of finite groups).
  
  The "retiring" in the file name refers to the office of AMS past president.
  
  Version of January 7, 2016.
  retiringHO.pdf
  Version aimed at undergrads, February 26, 2016.
  retiringENCHO.pdf
  
  

• Parameters for twisted representations
  Joint with Jeff Adams
  

  Writes down very concretely the data needed to specify a representation of a disconnected reductive group, in the form needed for the unitarity algorithm of the Adams-van Leeuwen-Trapa-Vogan paper below.
  
  51--116 in Representations of Reductive Groups, Monica Nevins and Peter Trapa, eds. Birkh\"auser (Springer) 2015.
  
  Version of August 15, 2015, 65 pages.
  twistedparams.pdf
  

• Errata for "Parameters for twisted representations"
  Joint with Jeff Adams
  

  A few of the more complicated formulas in the paper above turned out to contain sign errors: when implemented in the atlas software, they led to incorrect calculations of unitary representations. These notes were written while the software was being debugged. We are now (since February 2017) very confident that the software correctly computes unitary representations; what is questionable is whether I wrote down all of the sign errors we found.
  
  Version of January 14, 2017, 15 pages.
  errata.pdf
  

• Branching to maximal compact subgroups
  

  Slides for talk at Roger Howe 70th Birthday Conference at Yale, June 1-5 2015. Subject is an algorithm for computing the associated variety of an irreducible Harish-Chandra module.
  
  Version of June 3, 2015.
  howeHO.pdf
  
  Related talk Coherent sheaves on nilpotent cones June 22, 2015 and January 4, 2016, 20 pages.
  lusztig70HO.pdf
  

• Contragredient representations and characterizing the local Langlands correspondence
  Joint with Jeff Adams
  

  Addresses a question raised by Kevin Buzzard: what does the operation of taking contragredient look like on the level of Langlands parameters? In order to answer this question, we give a fairly concrete and elementary characterization of the Langlands classification. Lots of the results make sense for any local field, but proofs are complete only in the archimedean case.
  
  Amer. J. Math. 138 (2016), no. 3, 657-682.
  
  Version of March 26, 2015, 32 pages.
  contragredient.pdf
  

• Matrices almost of order two
  

  Slides for (first) a talk at the 40th Anniversary Midwest Representation Theory Conference, in honor of Becky Herb's 65th birthday, and in memory of Paul Sally, September 5-7, 2014. Topic is the local Langlands correspondence over the real numbers; mostly a restatement of the ideas in the book of Adams/Barbasch/Vogan.
  
  Version of September 5, 2014.
  almosttwoCHIC.pdf
  Version of September 19, 2014: more exposition, less technical detail.
  almosttwoOTT.pdf
  Version of September 24, 2014: more technical detail.
  almosttwoMITHO.pdf
  

• Quasisplit Hecke algebras and symmetric spaces
  Joint with George Lusztig
  

  Computes the action of an outer automorphism on the perverse cohomology spaces appearing in Kazhdan-Lusztig theory for real reductive groups. This information is used in the unitarity algorithm of the Adams-van Leeuwen-Trapa-Vogan paper below. below.
  
  Duke Math. J. 163 (2014), no. 9, 983-1034.
  
  Version of February 15, 2014, 47 pages.
  LV12arxH.pdf
  

• Unitary representations
  

  Slides for first three talks at the Workshop on Unitary Representation Theory, July 1-5, 2013.
  
  Version of July 3, 2013.
  atlas13CHO.pdf
  

• Understanding restriction to K
  

  Slides for a talk at the Representation Theory Conference in honor of Wilfried Schmid on the occasion of his 70th birthday. May 20-23, 2013. Topic is restriction to a maximal compact subgroup of irreducible representations of a real reductive group.
  
  Version of May 20, 2013.
  schmidHO.pdf
  Add example from Problem Session, correct minor typographical errors.
  schmidHOrevB.pdf
  

• Extended groups and representation theory
  

  Slides for a talk at the CUNY Representation Theory Seminar, April 19, 2013. Topic is the Adams-du Cloux description of real forms and their representations for complex reductive groups.
  
  Version of April 19, 2013.
  extendedHO.pdf
  Added page of root datum examples, minor edits.
  extendedHOrev.pdf
  

• Unitary representations of real reductive groups
  Joint with Jeff Adams, Marc van Leeuwen, and Peter Trapa
  

  Presents an algorithm for calculating the signature of any invariant Hermitian form on any irreducible representation of a linear real reductive Lie group, and (consequently) calculating the unitary dual. The main point (as in the work of Wai Ling Yee for category O) is to use results of Beilinson-Bernstein about the Jantzen filtration to relate signatures to Kazhdan-Lusztig polynomials. In the case when the Cartan involution is not inner, we need also to compute the action of the Cartan involution on the intersection cohomology spaces computed by KL polynomials. This was done in a Duke paper with Lusztig, available on the arxiv or above.
  
  Appeared in Asterisque 417.
  
  Version of January 20, 2013, 201 pages.
  hermitianForms.pdf
  Version of March 10, 2015, 168 pages.
  Corrected many small mistakes, including some sign errors in the main recursion theorems. Added example of SL(2,C).
  hermitianFormsSMF.pdf
  Corrected some typographical errors (thank you, Chaoping Dong!)
  hermitianFormsSMFcorr2.pdf
  Added some missing signs (transcription errors in applying lemmas) in the main theorems on pages 147-152.
  hermitianFormsSMFcorr6.pdf
  Version of November 1, 2019: revisions requested by referees. Down to 155 pages!
  hermitianFormsSMFrefC.pdf
  Version of January 15, 2022: as requested by Gregg Zuckerman, internal and bibliographical references are now links. 161 pages. This is close to the published version in numbering of theorems and references, but of course not in pagination.
  hermitianFormsSMFLINK.pdf
  

• Realizing smooth representations
  

  Slides for a talk at the Latin American Conference on Lie groups and Geometry at CIMAT, Guanajuato, in September, 2012. Concerns joint work in progress with David Jerison on the problem of finding something like eigenspace representations that can admit a unitary structure.
  
  Version of August 29, 2012.
  jerisonSLIDESHO.pdf
  

• Unitary representations of reductive groups
  

  Slides for ten lectures at the conference "CBMS 2012: Unitary Representations of Reductive Groups" at UMass Boston in July, 2012. The slides near the end (about the algorithm for computing signatures) are more or less copied from "Kazhdan-Lusztig polynomials for signatures" below. They were not actually delivered at the CBMS meeting, and the notation has not been made consistent with the rest of the slides.
  
  The July 21 version includes a few slides more or less covering what was done in Lecture 7 on the blackboard (a one-board summary of the book "Representations of real reductive Lie groups").
  
  Lectures 1-5, version of July 21, 2012 (tiny corrections from 7/20)
  cbms1.pdf
  Lectures 6-10, version of July 21, 2012
  cbms2.pdf
  

• Weyl group representations, nilpotent orbits, and the orbit method
  

  Slides for a talk at the conference "Lie groups: structure, action, representations" at Ruhr-Universität Bochum in January, 2012, on the occasion of Joe Wolf's 75th birthday. There is an outline of old work of Barbasch and others computing the relationships among the terms in the title, and some remarks on how to relate it to the new KL polynomials for involutions.
  
  Version of January 10, 2012.
  wolfHO.pdf
  

• Kazhdan-Lusztig polynomials for disconnected groups
  

  Slides for two twenty-minute talks the Boston AMS meeting on January 4, 2012. There is an outline of the role of classical KL polynomials in representation theory, and a sketch of some new polynomials (introduced with Lusztig) and the role that they will play.
  
  Revised January 22, 2012 to printable version.
  twistedHO.pdf
  

• Regular polyhedra and finite Coxeter groups
  

  Slides for a colloquium at Texas Tech on November 10, 2011, describing Coxeter's connection between the classification of regular polyhedra and finite Coxeter groups. The best parts are stolen from Bill Casselman's beautiful notes. I corrected and cleaned up the slides for a talk at Shandong University in Jinan on June 11, 2012; previous version remains as "regpolyOLD.pdf." Further slight edits December 2014; Shandong version is present as "regpolyMIDDLEAGED.pdf." Version for a colloquium at Tufts January 25, 2019 includes a discussion of the 120-cell and 600-cell in four dimensions. Some sign errors in the matrix calculation on page 16 were noticed by Sigurdur Helgason, and corrected in what is now here.
  
  Version of January 26, 2019.
  regpolyTUFTSHO2.pdf
  Version of December 11, 2014.
  regpoly.pdf
  Printable version (missing a few pictures of flags).
  regpolyHO.pdf
  

• Finite maximal tori
  Joint with Gang Han
  

  This is a draft of a paper attempting to extend the theory of root systems for compact groups by replacing the maximal torus with a finite maximal abelian subgroup (if one exists). There is no really satisfactory theory yet, but there are a great many beautiful examples. We will be submitting this paper around August, 2011; especially before that time, I would welcome comments or criticisms of the draft.
  
  Symmetry: representation theory and its applications, 269-303, Progr. Math. 257, Birkhauser/Springer, New York, 2014.
  
  Version of July 12, 2011, 36 pages.
  finitetori.pdf
  Version of April, 2013, 26 pages. Abstract, minor format changes.
  finitetoriAMS4.13.pdf
  Three typos corrected 10/14 (notably change 5 to 6 in Figure 3, page 23).
  finitetoriAMS10.14.pdf
  

• The size of infinite-dimensional representations
  

  Slides for a lecture given for the 2010 Nankai International Conference and Summer School on Representation Theory and Harmonic Analysis, Nankai University, Tianjin, China. Introduction to the notion of "Gelfand-Kirillov dimension" for infinite-dimensional representations. Interesting representations of a Lie group G often arise as spaces of functions on a manifold M. Such function spaces are usually infinite-dimensional. A fundamental problem is to start with an abstract representation of G, and to realize it as functions on some manifold M. A natural first step is to figure out what the dimension of M ought to be. Another way to say this is that we would like to determine the dimension of a manifold by looking at a vector space of functions on it. There is an immediate problem. Because all separable Hilbert spaces are topologically isomorphic, the space of square-integrable functions can tell us only whether M is finite or infinite. One might hope that a more subtle space like smooth functions could do better, but again we are disappointed: the spaces of smooth functions on infinite compact manifolds are all isomorphic as topological vector spaces.
  
  houHANDOUT.pdf
  hou.pdf
  

• The translation principle
  

  Slides for three lectures given for the 30th Winter School of Geometry and Physics, Srni, Czech Republic. Leisurely introduction to Zuckerman's "translation principle," which says that irreducible representations of real reductive groups must appear in families indexed by dominant weights.
  
  srniHO.pdf
  

• Infinite-dimensional representations of real reductive groups
  

  Notes for several lectures at a workshop of the atlas project. Goal is to introduce precisely the objects in the title, with emphasis on questions (like "strong real forms") that are particularly important for that project.
  
  Version of June 27, 2017, 62 pages.
  voganWorkshopJ.pdf
  

• Inflatable mathematics
  

  Slides for a lecture given for Sophus Lie Days at Cornell, April 27, 2009. Discussion aimed at undergraduates of the underlying ideas in the Schubert calculus, followed by an account of the E8 calculation with those ideas.
  
  cornellUND.pdf
  cornellUNDHO.pdf printable version
  metzC.pdf French version (with thanks to Salah Mehdi and Monica Nevins!)
  metzCHO.pdf printable French version
  UMAHO.pdf version delivered to MIT Undergraduate Math Association 11/20/20
  

• Kazhdan-Lusztig polynomials for signatures
  Joint with Jeffrey Adams, Marc van Leeuwen, Peter Trapa, and Wai Ling Yee

  Slides for a lecture given in the special session "Computational methods in Lie theory" at NCSU April 4-5, 2009. A very short outline of a conjectural algorithm for determining the unitary dual of a real reductive Lie group.
  
  Second file, from the Atlas conference at Utah 7/09, includes a more precise statement of the algorithm. (Slightly edited 7/28/09 to correct errors found by the audience.)
  
  Third file, from the Zuckerman 60th Birthday Conference at Yale 10/09 includes an exposition of how this is connected to the Zuckerman translation principle.
  
  Fourth file, from the VII Workshop on Lie Theory and Applications in Cordoba, includes a little introduction about why knowing all unitary representations is a good thing.
  
  Fifth file, from the CMS meeting in Windsor, Ontario in December, 2009, includes some additional expository material about SL(2) and about the Jantzen filtration.
  
  Sixth file, from four lectures given June 8-11, 2010 at the graduate summer school 2010 Nankai International Conference and Summer School on Representation Theory and Harmonic Analysis, Nankai University, Tianjin, China. Corrected, revised, and expanded 6/11/10.
  
  signatureraleighHO.pdf
  utahconfHO.pdf
  zuckermanHO.pdf
  cordoba.pdf
  windsorHO.pdf
  nankai10HO.pdf
  taipei12.10HO.pdf
  

• The orbit method for reductive groups

  These are slides for an exposition of the orbit method, given at the conference Lie theory and geometry: the mathematical legacy of Bertram Kostant, at the University of British Columbia in May, 2008. They are descendants of the Ritt lecture slides, but contain some additional material.
  
  kostantDV.pdf
  kostantHO.pdf (printable version, lacking a few bits of information on the display slides)
  

• Geometry and representations of reductive groups

  These are slides for an exposition of the orbit method, given as Ritt lectures at Columbia on December 13 and 14, 2007. (Still under construction.)
  
  rittC.pdf
  

• The character table for E₈

  This is an exposition of the calculation of the character table for the split real form of E₈ by the research group "Atlas of Lie groups and representations." There are brief explanations of the words in the preceding sentence, aimed at mathematicians not working in the field. There is also a very brief description of the mathematical basis of the calculation.
  
  Notices Amer. Math. Soc. 54 (2007), no. 9, 1122-1134.
  
  Version of November 21, 2007, 15 pages. (This incorporates corrections of some historical errors in the published version.)
  articleHIST.pdf
  

• Branching to a maximal compact subgroup

  This paper describes algorithms first to parametrize the irreducible representations of a maximal compact subgroup K (in a linear real reductive group G), and then to compute the restriction to K of a standard (infinite dimensional) representation of G. (Probably you know that the first problem was already solved by Cartan and Weyl. This is one of those papers where you'll come out at the end knowing quite a bit less than when you started.) Computer implementation of these algorithms is a goal of the atlas project.
  
  321-401 in Harmonic analysis, group representations, automorphic forms and invariant theory, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap. 12. World Sci. Publ., Hackensack, NJ, 2007.
  
  Version of June 1, 2007, 72 pages.
  khatHOWE2.pdf
  khatHOWE2.ps (postscript file)
  khatHOWE2.dvi
  

• Branching to maximal compact subgroups

  These are slides for a short lecture at Helgason's 80th birthday conference in Reykjavik (on August 15, 2007). The idea is to make a path from Helgason's theorem on which finite-dimensional representations are spherical, through Zuckerman's theorem on the restriction to K of a finite-dimensional representation, to the theorem of the paper above.
  
  Version of August 9, 2007, 110 pages. (Many pages are overlays; there are 17 complete pages.)
  helgason.pdf
  

• The character table for E₈

  These are slides for a public lecture at MIT (on March 19, 2007) on the calculation of the character table for the split real form of E₈ by the research group "Atlas of Lie groups and representations." The intended audience is MIT undergraduates, not necessarily in mathematics.
  
  The joint author space here ought to have a great many names in it, beginning with the nineteen members of the atlas group. (They're listed near the beginning of the slides.) There are lots of pretty pictures in the slides, almost all thanks to the efforts of other people: John Stembridge, Scott Crofts, and Wai Ling Yee come to mind immediately.
  
  Version of March 19, 2007, 224 pages. Edited April 17, 2013. (Many pages are overlays; a human might count 32 distinct pages.)
  E8TALKedit.pdf
  e8wpiHOedit.pdf printable version (missing a lot of graphics); also edited 4/17/13.
  audio file
  

• Errata for the book Cohomological Induction and Unitary Representations
  Joint with Anthony Knapp

  These corrections were prepared by Tony Knapp (thank you, Tony!). I tried to take the opportunity to post here the introduction to the book, but unfortunately AMSTeX has evolved enough in the last ten years that I can no longer make the styles files for the book work. For the enjoyment of the experts, I will include here a brief excerpt from the TeX file for the introduction:
  
  $G=SL(2,\bR)$
  
  Meanwhile Princeton University Press allows amazon.com to make images from entire books accessible and searchable; you can use this feature to locate all 327 pages containing the word "shall," for instance.
  
  Version of September 20, 2019, 3 pages.
  kv-corrections5.pdf
  

• Unitary Shimura correspondence for split real groups
  Joint with Jeffrey Adams, Dan Barbasch, Annegret Paul, and Peter Trapa

  This paper finds a relationship between complementary series representations for nonlinear coverings of split simple groups, and spherical complementary series for (different) linear groups. The main technique is Barbasch's method of calculating some intertwining operators purely in terms of the Weyl group.
  
  J. Amer. Math. Soc.. 20 (2007), no. 3, 701--751.
  
  Version of September 1, 2005, 52 pages.
  shimurav3.pdf
  shimurav3.ps (postscript file)
  shimurav3.dvi
  

• Unitary representations and complex analysis
  

  These are notes based on five lectures at the CIME summer school "Representation Theory and Complex Analysis" in Venice in June, 2004. The goal (not completely achieved) was to write down certain pre-unitary structures on group representations on Dolbeault cohomology spaces. The notes do describe the machinery necessary to formulate these questions. I will be very grateful to hear about errors, obscurities, and so on. Already I am grateful to several participants in the summer school for such assistance (and to many more for the pleasure of their company!).
  
  Representation Theory and Complex Analysis (CIME 2004), Andrea D'Agnolo, editor. Springer, 2008.
  
  Version of January 2, 2008, 86 pages. The minor revisions from 9/16/04 include some clarifications; additional reference for Conjecture 10.3 (thanks to Tim Bratten); and a couple of small typographical corrections. The manuscript was reset in LaTeX by Andrea D'Agnolo.
  veniceCORR.pdf
  veniceCORR.ps (postscript file)
  veniceCORR.dvi
  

• Three-dimensional subgroups and unitary representations
  

  This is the written version of a lecture at the conference "Mathematics and theoretical physics" held March 13-17, 2000 in Singapore. There are two topics: Dynkin's classification of homomorphisms of SU(2) into a compact Lie group, and the still unsolved problem of classifying spherical unitary representations of split groups over local fields. Arthur's conjecture connects these problems, and the goal is to see what light Dynkin's methods can shed on the unsolved one.
  
  Challenges for the 21st century (Singapore, 2000), 213-250, World Sci. Publ., River Edge, NJ, 2001.
  
  Version of July 27, 2000.
  sing.pdf
  sing.dvi
  sing.ps (postscript file)
  
  

• Isolated unitary representations
  

  This was written in 1992 as an appendix to a three-author paper that was never written. (I will leave to the experts the task of deducing the names of the three authors, and dividing blame equitably among them. As a hint, the authors represent four continents by birth and residence.) The main theorem says that Zuckerman's "A_q(lambda)" representations are isolated in the unitary dual, with a few obvious exceptions.
  
  Automorphic Forms and their Applications (2002), IAS/Park City Mathematics Series 12, 379-398. American Mathematical Society, Providence, RI (2007).
  
  Version of April 6, 2005.
  iso3.pdf
  iso3.dvi
  iso3.ps (postscript file)
  
  

• Unitary representations of reductive Lie groups
  

  These are the transparencies for a lecture at the conference "Mathematics towards the third millenium," held May 27-29, 1999 at the Accademia Nazionale dei Lincei in Rome. Essentially they are a telegraphic summary of the paper "The method of coadjoint orbits..." below. The introduction to the paper (corresponding to three of the transparencies) sketches an answer to the question "what is representation theory?" that is meant to be accessible to most mathematicians.
  
  Version of May 25, 1999.
  rome.pdf
  rome.dvi
  rome.ps (postscript file)
  
  Here is the manuscript of the paper, as published in
  
  Rend. Mat. Acc. Lincei, 9 (2000), 147-167.
  
  Version of July 12, 1999.
  romems.pdf
  romems.dvi
  romems.ps (postscript file)
  
  

• The method of coadjoint orbits for real reductive groups
  

  These are notes for lectures at the Graduate Summer School in Representation Theory in Park City in July, 1998. In addition to general nonsense on the title subject, there is a brief account of some new ideas about quantization for nilpotent orbits.
  
  Representation Theory of Lie Groups, IAS/Park City Mathematics Series 8 (1999), 179-238.
  
  Version of September 30, 1998; corrects many typos throughout, and several obscurities in the last two lectures. (Thank you, Monica!)
  PCorb.pdf
  PCorb.dvi
  PCorb.ps (postscript file)
  
  

• A Langlands classification for unitary representations
  

  This is an expository account of the ideas in the following paper with Salamanca-Riba.
  
  Analysis on homogeneous spaces and representation theory of Lie groups, Okayama-Kyoto (1997), 299-324, Adv. Stud. Pure Math., 26, Math. Soc. Japan, Tokyo, 2000.
  
  Version of April 2, 1998.
  kyoto.pdf
  kyoto.dvi
  kyoto.ps (postscript file)
  
  

• On the classification of unitary representations of reductive Lie groups
  Joint with Susana Salamanca-Riba

  The goal is to understand the easy part of the role of cohomological induction in the classification of unitary representations.
  
  Annals of Mathematics 148 (1998), 1067-1133.
  
  Version of December 16, 1997.
  unitAMrev.pdf
  unitAMrev.dvi
  unitAMrev.ps (postscript file)
  
  

• Functions on the model orbit in E8
  Joint with Jeffrey Adams and Jing-Song Huang

  We do a calculation about representations of algebraic groups that has some conjectural meaning for infinite-dimensional representation theory.
  
  Representation Theory 2 (1998), 224-263.
  
  Version of April 17, 1998.
  model.pdf
  model.dvi
  model.ps (postscript file)
  
  

• Cohomology and group representations
  

  This is an expository paper about continuous cohomology for unitary representations of real reductive groups.
  
  Representation Theory and Automorphic Forms (Instructional Conference, International Centre for Mathematical Sciences, Edinburgh, March, 1996), T. Bailey and A. Knapp, editors. Proceedings of Symposia in Pure Mathematics 61. American Mathematical Society, Providence, RI (1997).
  
  Version of December 23, 1996.
  cohorev.pdf
  cohorev.dvi
  cohorev.ps (postscript file)
  
  

• Geometric quantization for nilpotent coadjoint orbits
  Joint with William Graham

  We look at the problem of attaching a representation to a nilpotent coadjoint orbit of a real reductive Lie group. The Kirillov-Kostant strategy of finding an invariant Lagrangian foliation of the orbit often cannot succeed in this case. We follow instead an idea of Guillemin-Sternberg and Ginsburg, working with a larger invariant family of Lagrangian submanifolds.
  
  Geometry and Representation Theory of real and p-adic groups. Birkhauser, Boston-Basel-Berlin, 1998 .
  
  Version of April 22, 1996
  quant.pdf
  quant.dvi
  quant.ps (postscript file)
  
  

• The orbit method and unitary representations for reductive Lie groups
  

  This is an expository paper.
  
  Algebraic and Analytic Methods in Representation Theory (Sonderborg, 1994). Perspectives in Mathematics 17. Academic Press, San Diego 1997.
  
  Version of December 22, 1994
  dmkrev.pdf
  dmkrev.dvi
  dmkrev.ps (postscript file)
  
  

• The local Langlands conjecture
  

  This is a draft of an exposition of formal aspects of formulating Kazhdan-Lusztig conjectures and Arthur's conjectures for p-adic reductive groups. The final version may be found in
  
  Representation Theory of Groups and Algebras (J. Adams et al., eds. Contemporary Mathematics 145. American Mathematical Society, 1993.
  
  Version of August 10, 1992
  md.pdf
  md.dvi
  md.ps
  
  
  

• The Langlands classification and irreducible characters (introduction).
  Joint with Jeffrey Adams and Dan Barbasch

  This is the introduction to a book explaining how to formulate and prove Kazhdan-Lusztig conjectures and Arthur's conjectures for real reductive algebraic groups. (Well, all of Arthur's conjectures except for the interesting parts, which say that certain representations are unitary.) If you're still awake at the end, the full text is in
  
  The Langlands Classification and Irreducible Characters for Real Reductive Groups (J. Adams, D. Barbasch, and D. Vogan). Progress in Mathematics 104. Birkhauser, Boston-Basel-Berlin, 1992.
  
  Version of April 8, 1992
  abvINT.pdf
  abvINT.dvi
  abvINT.ps
  
  
  

• Arthur packets and unitary representations

  This is streaming video of a one-hour lecture at Arthur's 60th birthday conference in Toronto. The lecture is meant to be motivation for the book The Langlands classification and irreducible characters. Unfortunately the video ends a minute or so before the lecture did, so we never learn whether the Mounties were able to apprehend the villain. (They were.) What is here is still a reasonable introduction to the introduction above.
  
  Version of October 15, 2004
  Fields Institute video
  
  
  

• Associated varieties and unipotent representations
  

  The point of this paper is to look at the relationship between an irreducible representation and its associated variety, particularly with a view to understanding the orbit method better. The paper appears in
  
  Harmonic Analysis on Reductive Groups (W. Barker and P. Sally, eds). Progress in Mathematics 101. Birkhauser, 1991.
  
  I have lost the original TeX file, and got tired of not having electronic access to the paper, so scanned it. Apologies for the resulting large file size.
  
  Scanned May 9, 2013
  assocvarunip.pdf (3 megabytes)
  
  
  

• Dixmier algebras, sheets, and representations
  

  This paper examines the relationship between induction (following Dixmier) of primitive ideals and induction (following Lusztig-Spaltenstein) of nilpotent orbits. The paper appears in
  
  Operator algebras, unitary representations, enveloping algebras, and invariant theory (A. Connes, M. Duflo, A. Joseph, and R. Rentschler, eds). Progress in Mathematics 92. Birkhauser, 1990.
  
  I typed this paper in a Commodore 64 word processor called PaperClip (later stolen I think by Microsoft) but I have neither the file nor the software nor the hardware to print it; so this is scanned. (Ann Kostant had it TeXed for Birkhauser, and she may have a TeX file.) Apologies for the large file size.
  
  Scanned January 24, 2017
  DixmierAlgebras.pdf (2.3 megabytes)
  
  
  

• The orbit method and primitive ideals for semisimple Lie algebras
  

  This paper examines what Kostant's ideas have to say about primitive ideals. One issue is how bad/interesting behavior of the orbits gives rise to bad/interesting behavior of primitive ideals. The paper appears in
  
  Lie algebras and related topics, AMS/CMS 1986.
  
  Like the one above, this paper was typed in PaperClip but I have neither the file nor the software nor the hardware to print it; so this is scanned. Apologies for the large file size. I made the bitmaps for the Gothic characters myself; for this also I apologize.
  
  Scanned January 28, 2019
  vogan86CMS.pdf (26.6 megabytes)
  
  
  

• Representations of Real Reductive Lie Groups (Introduction and Chapter 1).
  

  The topic of this book is the construction and classification of all irreducible representations of real reductive Lie groups, using ideas introduced by Zuckerman in the late 1970s. The topic of Chapter 1 is the special case of SL(2,R). Thanks to Wai Ling Yee for preparing the scan. (The book is no longer in print, but work on a second edition is in progress.)
  
  Representations of Real Reductive Groups. Progress in Mathematics 15. Birkhauser, Boston-Basel-Berlin, 1981.
  
  Scan of 1981 book
  vogan81chapter1.pdf
  
  
  

• Lie algebra cohomology and the representations of semisimple Lie groups
  

  This is my dissertation, written under the direction of Bert Kostant. The mathematical content is mostly done better and more generally in the book above; the dissertation includes many case-by-case calculations, and also a discussion of nonlinear groups.
  
  Lie algebra cohomology and the representations of semisimple Lie groups, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1976.
  
  Scan of 1976 dissertation
  vogan76.pdf (40 megabytes)
  
  
  

Return to David Vogan's homepage