Papers by David Vogan (...and his friends)
Because I like the color red, and because I'm not writing so many
papers these days, I will try to introduce a color coding scheme:
red denotes slides for lectures,
blue denotes manuscripts of papers, and black denotes books.
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.
Laplacians on spheres
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.
Joint with Henrik Schlichtkrull and Peter Trapa
Version of March 3, 2018, 49 pages.
The size of infinite-dimensional representations
Based on Takagi Lectures below.
Version of August 2, 2017 (small corrections), 35 pages.
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.
Translation principle for Dirac
Relates the index of the Dirac operator (for
real reductive group representations) to multiplicities of certain
orbits in the associated cycle of the representation.
Joint with Salah Mehdi and Pavle Pandzic
Version of April 2, 2016, 23 pages.
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.
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; I don't yet have plans to retire from MIT.
Version of January 7, 2016.
Version aimed at undergrads, February 26, 2016.
Parameters for twisted
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
Joint with Jeff Adams
51--116 in Representations of Reductive Groups, Monica
Nevins and Peter Trapa, eds. Birkh\"auser (Springer) 2015.
Version of August 15, 2015, 65 pages.
Errata for "Parameters for twisted
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.
Joint with Jeff Adams
Version of January 14, 2017, 15 pages.
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
Version of June 3, 2015.
Related talk Coherent sheaves on nilpotent cones June 22, 2015 and January 4, 2016, 20 pages.
Contragredient representations and characterizing the local
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.
Joint with Jeff Adams
Version of March 26, 2015, 32 pages.
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
Version of September 5, 2014.
Version of September 19, 2014: more exposition, less technical detail.
Version of September 24, 2014: more technical detail.
Quasisplit Hecke algebras and symmetric spaces
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.
Joint with George Lusztig
Duke Math. J. 163 (2014), no. 9,
Version of February 15, 2014, 47 pages.
Unitary representations Slides
for first three talks at the Workshop on Unitary Representation
Theory, July 1-5, 2013.
Version of July 3, 2013.
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
Version of May 20, 2013.
Add example from Problem Session, correct minor typographical errors.
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.
Added page of root datum examples, minor edits.
Unitary representations of real reductive
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.
Joint with Jeff Adams, Marc van Leeuwen, and Peter
Version of January 20, 2013, 201 pages.
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).
Corrected some typographical errors (thank you, Chaoping Dong!)
Added some missing signs (transcription errors in applying lemmas) in the main theorems on pages 147-152.
Version of July 19, 2017: revisions requested by referees. Down to 154 pages!
Realizing smooth representations Slides for a talk at the
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.
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)
Lectures 6-10, version of July 21, 2012
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
Version of January 10, 2012.
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.
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
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 of December 11, 2014.
Printable version (missing a few pictures of flags).
Finite maximal tori
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.
Version of April, 2013, 26 pages. Abstract, minor format changes.
Three typos corrected 10/14 (notably change 5 to 6 in Figure 3, page 23).
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.
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.
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.
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.
cornellUNDHO.pdf printable version
metzC.pdf French version (with thanks to Salah Mehdi and Monica Nevins!)
metzCHO.pdf printable French version
Kazhdan-Lusztig polynomials for signatures 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.
Joint with Jeffrey
Adams, Marc van Leeuwen, Peter Trapa, and Wai Ling Yee
Second file, from
conference at Utah 7/09, includes a more precise statement of the
algorithm. (Slightly edited 7/28/09 to correct errors found by the
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
Workshop on Lie Theory and Applications in Cordoba, includes a little
introduction about why knowing all unitary representations is a good
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.
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
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
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,
Version of November 21, 2007, 15 pages. (This incorporates corrections
of some historical errors in the published version.)
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.ps (postscript file)
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.)
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
Version of March 19, 2007, 224 pages. Edited April 17, 2013. (Many pages are overlays; a
human might count 32 distinct pages.)
e8wpiHOedit.pdf printable version (missing a
lot of graphics); also edited 4/17/13.
Errata for the book Cohomological Induction and Unitary
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:
Joint with Anthony Knapp
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
Version of August 23, 2009, 3 pages.
Unitary Shimura correspondence for split real groups
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.
Joint with Jeffrey Adams, Dan Barbasch, Annegret Paul, and Peter Trapa
J. Amer. Math. Soc.. 20 (2007), no. 3,
Version of September 1, 2005, 52 pages.
shimurav3.ps (postscript file)
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.ps (postscript file)
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.
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
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.
Unitary representations of reductive Lie groups
These are the transparencies for a lecture at
"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
Version of May 25, 1999.
Here is the manuscript of the paper, as published in
Rend. Mat. Acc. Lincei, 9 (2000), 147-167.
Version of July 12, 1999.
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
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!)
A Langlands classification for unitary representations
This is an expository account of the ideas in the following paper with
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.
On the classification of unitary representations
of reductive Lie groups
The goal is to understand the easy part of the role of
cohomological induction in the classification of unitary
Joint with Susana Salamanca-Riba
Annals of Mathematics 148 (1998), 1067-1133.
Version of December 16, 1997.
Functions on the model orbit in E8
We do a calculation about representations of algebraic groups that has
some conjectural meaning for infinite-dimensional representation
Joint with Jeffrey Adams and Jing-Song Huang
Representation Theory 2 (1998), 224-263.
Version of April 17, 1998.
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.
Geometric quantization for nilpotent coadjoint orbits
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.
Joint with William Graham
Geometry and Representation Theory of real and p-adic
groups. Birkhauser, Boston-Basel-Berlin, 1998 .
Version of April 22, 1996
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
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
The Langlands classification and irreducible characters
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
Joint with Jeffrey Adams and Dan Barbasch
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,
Version of April 8, 1992
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
Associated varieties and unipotent
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
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
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)
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
Representations of Real Reductive Groups. Progress
in Mathematics 15. Birkhauser, Boston-Basel-Berlin,
Scan of 1981 book
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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