De-Rham theorem and Shapiro lemma for Schwartz functions on Nash manifolds

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 de-Rham theorem for de-Rham complexes with coefficients in Schwartz functions and generalized Schwartz functions. Using that we compute cohomologies of the Lie algebra 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 Shapiro lemma.

21 pages, v2: minor corrections

References in corpus (1)

• Schwartz functions on Nash manifolds

Cited by in corpus (5)

• Twisted homology for the mirabolic nilradical
• Smooth Transfer (the Archimedean case)
• Hausdorffness for Lie algebra homology of Schwartz spaces and applications to the comparison conjecture
• Generalized Harish-Chandra descent and applications to Gelfand pairs
• Multiplicity One Theorems and Invariant Distributions