The Unipotent Muffin Research Kitchen
Here you'll find a PostgreSQL database
that contains various structural data for the simple complex Lie algebras, their real forms,
their root subsystems, their Weyl group representations, and their nilpotent orbits. The data
set is complete up to rank 8. Data in the individual tables can be accessed by filling out a
simple SQL query form. Questions about the correlations amongst multiple tables can answered by
submitting an appropriate query in SQL, which is both a straightforward
and standard language for accessing data in a relational database.
Regular users of the UMRK database may find it efficient to have a local
mirror of the UMRK setup. Such a mirror could be set up a couple of ways.
VirtualBox is a free Oracle virtual machine software that runs on Windows, Linux, Apple's OS X,
and Solaris hosts. Once VirtualBox is installed on a local machine, a
UMRK virtual machine can be run using the VM (virtual machine) files in
http://umrkportal.net/~binegar/UMRK/VirtualBoxVM. Once started,
the URMK VM will need to obtain a ip address via DHCP; and once it connects
itself to the local network, it will accept and answer web browser queries
just like the original UMRK machine.
Alternatively, if one has an SQL database program already installed on a local
machine, one could reconstruct the URMK data tables using the SQL command
files found in
http://umrkportal.net/~binegar/UMRK/SQL_files.
Finally, the UMRK data tables are available in the form of CSV
(colon delimited text) files. See
http://umrkportal.net/~binegar/UMRK/tables. These CSV files
could then be, for example, imported into an Excel spreadsheet.
Tables of Atlas and Lie Data
Here you'll find
data files produced by the software program Atlas as well as data files
for things like Green Polynomials, Primitive Ideals, KL polynomials,
Duflo involutions, etc. See the text file 00-Readme.txt or the text file
00-TableOFContents (in the top level directory for a given Cartan type)
for a brief listing of the available data.
Harish-Chandra Cells, Nilpotent Orbits, Primitive Ideals and Weyl Group Representations
Here you'll find tables that,
for each real form of the complex exceptional Lie groups, detail the organization of W-cells of
the irreducible admissible representations of regular integral
infinitesimal character in terms of nilpotent orbits, primitive ideals, and Weyl groups representations.
Representations of Lie Groups and Applications, Institut Henri
Poincare, Paris, December 15-18, 2008
Here you can find the slides from my talk
HC-Cells, Nilpotent Orbits, Primitive
Ideals and Weyl Group Representations - slides.
And here you can find a version of my talk in prose form, with references and the tables I mentioned at the end of my talk.
HC-Cells, Nilpotent Orbits, Primitive Ideals, and Weyl Group Representations - text
2008 Nankai Summer School in Representation Theory and
Harmonic Analysis, Nankai University, June 2008
In June of 2008 I gave a series of eight lectures on
computational aspects of representation theory and the Atlas
for Lie Groups project. See
this link
for details. Notes for these lectures can be found
here.
Current/Recent Research Papers
On a class of multiplicity-free K_C orbits
(J. Math. Kyoto Univ. Vol 47, No. 4 (2007),735-766)
Abstract:
Let G be a real, connected, noncompact, semisimple Lie group,
let K_C be the complexification of a maximal compact subgroup K of G, and
let g = k+p be
the corresponding Cartan decomposition of the complexified Lie algebra of G.
Sequences of strongly orthogonal noncompact weights are constructed and
classified for each real noncompact simple Lie group of classical type. We
show that for each partial subsequence gamma_{1},...,gamma_{i} there is a
corresponding family of nilpotent K_C orbits
in p, ordered by
inclusion and such that the representation of K on the ring of regular
functions on each orbit is multiplicity-free. The K-types of regular functions
on the orbits and the regular functions on their closures are both
explicitly identified and demonstrated to coincide,
with one exception in the Hermitian symmetric case.
The classification presented also includes
the specification of a base point for each orbit and exhibits a corresponding
system of restricted roots with multiplicities. A formula for the
leading term of the Hilbert polynomials corresponding to these orbits is given.
This formula, together with the restricted root data, allows the determination
of the dimensions of these orbits and the algebraic-geometric degree of their
closures. In an appendix, the location of these orbits within D. King's
classification of spherical nilpotent orbits in complex symmetric spaces
is depicted via signed partitions and Hasse diagrams.
On the evaluation of some Selberg-like integrals
(J. of Math. Anal. Appl. 343 (2008) 601-620 )
Abstract:
Several methods of evaluation are presented for a family
of Selberg-like
integrals that arise in the computation of the algebraic-geometric
degrees of a family of spherical nilpotent orbits associated to the
symmetric space of a simple real Lie group. Adapting the technique of
Nishiyama, Ochiai and Zhu, we present an explicit evaluation in terms of
certain iterated sums over permutation groups. The resulting formula, however,
is only valid when the integrand involves an even power of the Vandermonde
determinant. We then apply, to the general case, the theory of symmetric functions and obtain an evaluation of the integral Indp as a product of
polynomial of fixed degree times a particular product of gamma factors;
thereby identifying the asymptotics of the integrals with respect to
their parameters. Lastly, we derive a recursive formula for evaluation
of another general class of Selberg-like integrals, by applying some of
the technology of generalized hypergeometric functions.
Recent Talks
Nilpotent Orbits and Weyl
Group Represenations - OSU Lie Groups Seminar, April 2017
Variations on a Formula of Barbasch and Vogan,
Slides from a talk given at the Special Session on Lie Groups and Automorphic
Forms, CMS Winter Meeting, Windsor, Canada, December 6, 2009.
Subsystems, Nilpotent Orbits and Weyl Group Representations,
notes from a talk given at the OSU Lie Groups Seminar, November 18, 2009.
A combinatorial parameterization of nilpotent orbits, twisted induction and duality, I,
notes from a talk given at the OSU Lie Groups Seminar, February 27,
2008.
A combinatorial parameterization of nilpotent orbits, twisted induction and duality, II,
notes from a talk given at the OSU Lie Groups Seminar, March 5,
2008.
Tau signatures, orbits and cells, I,
notes from a talk given at the OSU Lie Groups Seminar, November 28,
2007.
Tau signatures, orbits and cells, II,
notes from a talk given at the OSU Lie Groups Seminar, December 12,
2007.
A taxonomy of the irreducible Harish-Chandra modules of regular integral infinitesimal character, I,
notes from a talk given at the OSU Lie Groups Seminar, September 2007.
A taxonomy of the irreducible Harish-Chandra modules of regular integral infinitesimal character, II,
notes from a talk given at the OSU Lie Groups Seminar, September 2007.
A fine partitioning of cells,
notes from a talk given at the Atlas Workshop, AIM, July 2007.
Whittaker vectors, a matrix
calculus, and generalized hypergeometric functions, slides from a talk
given at the International Symposium on Representation Theory, Systems of
Differential Equations and Related Topics, Hokkaido University, July 2007.
On a family of spherical nilpotent
K_C orbits, talk given at Atlas Workshop, MIT, March 2007.
Spherical Nilpotent Orbits
and Unipotent Representations
OSU Representation Theory Seminar, November 8, 2006
On a class of multiplicity-free
K_C orbits
OSU Representation Theory Seminar, May 5, 2006
Berstein degree computations and
Selberg type integrals
OSU Representation Theory Seminar, March 1, 2006
Whittaker vectors,
generalized hypergeometric functions and a matrix calculus
Lie Groups Workshop, UCSB, February 4, 2006
Variations on a formula of Selberg, I
OSU Representation Theory Seminar, September, 2005
Variations on a formula of Selberg, II
OSU Representation Theory Seminar, September, 2005
Characteristic cycles, multiplicities
and degrees for a class of small unitary representations, I
OSU Representation Theory Seminar, March 2004
Characteristic cycles, multiplicities
and degrees for a class of small unitary representations, II
OSU Representation Theory Seminar, March 2004
Shared Orbits
OSU Representation Theory Seminar, November, 2003