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On the role of effective representations of Lie groupoids. (English) Zbl 1216.58005

The Tannaka duality construction for the ordinary representations of a proper Lie groupoid \(\mathcal G\) on vector bundles is studied. The author shows that (Theorem 2.5) the canonical homomorphism \(\pi_{\mathcal G}\) of \(\mathcal G\) into the reconstructed groupoid \(\mathcal T(\mathcal G)\) called Tannakian bidual is surjective. The author uses a notion of smooth structure by a structured space endowed with a sheaf of real algebras of continuous real valued functions and proves that (Theorem 2.9) \(\pi_{\mathcal G}\) is an isomorphism of smooth groupoids if and only if, for each base point \(x\), the induced isotropy homomorphism \(\rho_x: \mathcal G_x\to GL(E_x)\) is trivial, where \(\rho: \mathcal G\to GL(E)\) is a representation of \(\mathcal G\) on the complex vector bundle \(E\) over \(\mathcal G^{(0)}\). The author calls an isotropic arrow \(g\in \mathcal G_x\) ineffective if the action of \(g\) on the normal space to the \(\mathcal G\)-orbit at \(x\) is trivial, and characterizes a proper Lie groupoid whose bidual is a Lie groupoid, by the following condition (Theorem 4.10): For each base point \(x\), there is a representation such that the kernel of the corresponding isotropy homomorphism \(\rho_x\) is injective, i.e., sits inside the ineffective subgroup of \(\mathcal G_x\). This condition is examined in detail and some counterexamples are mentioned.

MSC:

58H05 Pseudogroups and differentiable groupoids
22D35 Duality theorems for locally compact groups
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