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

• 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
  

• 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.
  
  Version of June 5, 2007, 15 pages.
  notices07.pdf
  notices07.ps (postscript file)
  notices07.dvi
  

• 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.
  
  To appear in Harmonic Analysis, Group Representations, Automorphic Forms and Invariant Theory: In Honour of Roger E. Howe, edited by Jian-Shu Li, Eng-Chye Tan, Nolan Wallach and Chen-Bo Zhu, Singapore University Press and World Scientific Publishing.
  
  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
  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, 2005, 2 pages.
  KVcorr05.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
  
  
  

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