Fredholm conditions for operators invariant with respect to compact Lie\n group actions

2020/12/07 by Alexandre Baldare, Baldare, Alexandre, Rémi Côme +3
Mathematics · #Advanced Operator Algebra Research #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Representation Theory (math.RT) #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2012.03944

openalex publication_date 2020/12/07 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

Abstract

Let G be a compact Lie group acting smoothly on a smooth, compact manifold\nM, let P \∈ \ψ^m(M; E₀, E₁) be a G--invariant, classical\npseudodifferential operator acting between sections of two vector bundles E_i\n\→ M, i = 0,1, and let \α be an irreducible representation of the\ngroup G. Then P induces a map \π_\α(P) : H^s(M; E₀)_\α \→\nH^s-m(M; E₁)_\α between the \α-isotypical components. We prove\nthat the map \π_\α(P) is Fredholm if, and only if, P is em\ntransversally \α-elliptic, a condition defined in terms of the principal\nsymbol of P and the action of G on the vector bundles E_i.\n

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