On covariant functions and distributions under the action of a compact group

2009/02/09 by Anouar Saidi, Saidi, Anouar
Mathematics · #46F05 #58C46. #58C81 #58C99 #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Holomorphic and Operator Theory #Representation Theory (math.RT) #math.RT #msc:46F05 #msc:58C46. #msc:58C81 #msc:58C99

paper · pdf · doi:10.48550/arxiv.0902.1383

arxiv created 2009/02/09 · openalex publication_date 2009/02/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a compact subgroup of GL_n(\R) acting linearly on a finite dimensional vector space E. B. Malgrange has shown that the space C^∞(\Rⁿ,E)^G of C^∞ and G-covariant functions is a finite module over the ring C^∞(\Rⁿ)^G of C^∞ and G-invariant functions. First, we generalize this result for the Schwartz space \mathscrS(\Rⁿ,E)^G of G-covariant functions. Secondly, we prove that any G-covariant distribution can be decomposed into a sum of G-invariant distributions multiplied with a fixed family of G-covariant polynomials. This gives a generalization of an Oksak result proved in ([O]).

Related

• Distributions on homogeneous spaces and applications
• Homogeneous distributions on finite dimensional vector spaces
• Orbit misbehavior, isotropy discontinuity, and large isotypic components
• Distributions invariantes sur les groupes reductifs quasi-deployes
• First Fundamental Theorem of Invariant Theory for covariants of classical groups