2008 Nankai Summer School in Representation Theory and
Harmonic Analysis
In June of 2008 I will be giving a series of seven lectures on
computational aspects of representation theory and the Atlas
for Lie Groups project. See
this link
for details.
Current/Recent Research Papers
On a class of multiplicity-free K_C orbits
(accepted for publication by J. of Math. of Kyoto Univ.)
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 (accepted for
publication by Journal of Mathematical Analysis and Applications)
Abstract:
Several methods of evaluation are presented for a family
{Indp} 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
Tau signatures, orbits and cells, I,
notes from a talk given at the OSU Lie Groups Seminar Workshop, November 28,
2007.
Tau signatures, orbits and cells, II,
notes from a talk given at the OSU Lie Groups Seminar Workshop, 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 Workshop, 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 Workshop, 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