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

• 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.
  
  Version of November 11, 2011.
  regpoly.pdf
  Printable version (missing a few pictures of flags) January 22, 2012.
  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.
  
  Version of July 12, 2011, 36 pages.
  finitetori.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
  

• 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
  

• 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. (Many pages are overlays; a human might count 32 distinct pages.)
  E8TALK.pdf
  e8wpiHO.pdf printable version (missing a lot of graphics)
  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 August 23, 2009, 3 pages.
  kv-corrections4.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
  abv.pdf
  abv.dvi
  abv.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
  
  
  

• 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)
  
  
  

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