Abstract
In this paper we continue our work on Schwartz functions and generalized Schwartz functions on Nash (i.e. smooth semi-algebraic) manifolds. Our first goal is to prove analogs of the de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute the cohomologies of the Lie algebra g of an algebraic group G with coefficients in the space of generalized Schwartz sections of G-equivariant bundle over a G-transitive variety M. We do it under some assumptions on topological properties of G and M. This computation for the classical case is known as the Shapiro lemma.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Aizenbud and D. Gourevitch, Schwartz functions on Nash manifolds, International Mathematics Research Notices IMRN no. 5 (2008). Art. ID rnm 155, 37 pp. See also arXiv:0704.2891v3 [math.AG].
J. Bochnak, M. Coste and M.-F. Roy, Real Algebraic Geometry, Berlin, Springer, 1998.
R. Bott and L. Tu, Differential Forms in Algebraic Topology, New York, Springer, 1982.
W. Casselman, H. Hecht and D. Miličić, Bruhat filtrations and Whittaker vectors for real groups, in The Mathematical Legacy of Harish-Chandra (Baltimore, MD, 1998), Proceedings of Symposia in Pure Mathematics, 68, American Mathematical Society, Providence, RI, 2000, pp. 151–190.
P. Deligne, Thèorie de Hodge. III, Publications Mathématiques. Institut de Hautes Études Scientifiques 44 (1974), 5–77.
H. Delfs and M. Knebush, Semialgebraic topology over a real closed field II, Mathematische Zeitschrift 178 (1981), 175–213.
H. Hironaka, Resolution of singularities of an algebraic variety over a field of characteristic zero, I, II, Annals of Mathematics 79 (1964), 109–326.
B. Malgrange, Ideals of Differentiable Functions, Oxford University Press, Oxford, 1966.
W. Rudin, Functional analysis, McGraw-Hill, New York, 1973.
M. Shiota, Nash Manifolds, Lecture Notes in Mathematics 1269, Springer, New York, 1987.
F. W. Warner, Foundation of Differentiable Manifolds and Lie Groups, Springer, New York, 1983.
Author information
Authors and Affiliations
Corresponding author
Additional information
Both authors were partially supported by a BSF grant, a GIF grant, and an ISF Center of excellency grant.
A. Aizenbud was also supported by ISF grant No. 583/09.
Rights and permissions
About this article
Cite this article
Aizenbud, A., Gourevitch, D. The de-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds. Isr. J. Math. 177, 155–188 (2010). https://doi.org/10.1007/s11856-010-0042-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-010-0042-9