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.

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