
SIGMA 4 (2008), 069, 33 pages arXiv:0801.3277
https://doi.org/10.3842/SIGMA.2008.069
Contribution to the Special Issue on KacMoody Algebras and Applications
Homogeneous Poisson Structures on Loop Spaces of Symmetric Spaces
Doug Pickrell
Department of Mathematics, University of Arizona, Tucson,
AZ, 85721, USA
Received June 14, 2008, in final form September 27, 2008; Published online October 07, 2008
Abstract
This paper is a sequel to [Caine A., Pickrell D., Int. Math. Res. Not., to appear, arXiv:0710.4484], where we
studied the Hamiltonian systems which arise from the EvensLu
construction of homogeneous Poisson structures on both compact and
noncompact type symmetric spaces. In this paper we consider loop
space analogues. Many of the results extend in a
relatively routine way to the loop space setting, but new issues
emerge. The main point of this paper is to spell out the meaning
of the results, especially in the SU(2) case. Applications
include integral formulas and factorizations for Toeplitz
determinants.
Key words:
Poisson structure; loop space; symmetric space; Toeplitz determinant.
pdf (396 kb)
ps (261 kb)
tex (34 kb)
References
 Brezis H., New questions related to
topological degree, in The Unity of Mathematics, Prog. Math., Vol. 244, Birkhäuser, Boston, MA, 2006, 137154.
 Caine A., Compact symmetric spaces, triangular
factorization, and Poisson geometry, J. Lie Theory 18 (2008),
273294, math.SG/0608454.
 Caine A., Pickrell D., Homogeneous Poisson structures on symmetric
spaces, Int. Math. Res. Not., to appear, arXiv:0710.4484.
 Evens S., Lu J.H., On the variety of Lagrangian subalgebras. I, Ann. Sci. École Norm. Sup. (4) 34 (2001), 631668, math.DG/9909005.
 Kac V., Infinitedimensional Lie algebras. An introduction,
Birkhäuser, Boston, MA, 1983.
 Kac V., Constructing groups from infinitedimensional Lie algebras, in InfiniteDimensional Groups with
Applications (Berkeley, Calif., 1984), Editor V. Kac, Math. Sci. Res. Inst. Publ., Vol. 4, Springer, New York, 1985, 167216.
 Lu J.H., Coordinates on Schubert cells,
Kostant's harmonic forms, and the BruhatPoisson structure on
G/B, Transform. Groups 4 (1999), 355374, dgga/9610009.
 Odzijewicz A., Ratiu T., Banach
LiePoisson spaces and reduction, Comm. Math. Phys. 243
(2003), 154, math.SG/0210207.
 Pickrell D.,
An invariant measure for the loop space of a simply
connected compact symmetric space, J. Funct. Anal. 234 (2006),
321363, mathph/0409013.
 Pickrell D., A survey of conformally
invariant measures on H^{m}(D), math.PR/0702672.
 Pressley A., Segal G., Loop groups, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986.
 Widom H., Asymptotic behavior of block
Toeplitz matrices and determinants. II, Adv. Math. 21 (1976),
129.

