The equivariant coarse index is well-understood and widely used for actions by discrete groups. We first extend the definition of this index to general locally compact groups. We use a suitable notion of admissible modules over C *algebras of continuous functions to obtain a meaningful index. Inspired by a work of Roe, we then develop a localised variant, with values in the K-theory of a group C * -algebra. This generalises the Baum-Connes assembly map to non-cocompact actions. We show that an equivariant index for Callias-type operators is a special case of this localised index, obtain results on existence and non-existence of Riemannian metrics of positive scalar curvature invariant under proper group actions, and show that a localised version of the Baum-Connes conjecture is weaker than the original conjecture, while still giving a conceptual description of the K-theory of a group C * -algebra.Résumé. -L'indice grossier équivariant est bien compris et utilisé pour les actions par les groupes discrets. On commence par étendre la définition de cet indice aux groupes localement compacts généraux. On utilise une notion de modules admissibles sur des C * -algèbres de functions continues, pour obtenir un indice utile. Inspirés par le travail de Roe, nous développons une variante localisée, à valeurs dans la K-théorie de la C * -algèbre d'un groupe, généralisant l'assembly map de Baum-Connes aux actions non-cocompactes. On montre qu'un indice pour des opérateurs de type Callias est un cas spécial de cet indice localisé; on obtient des résultats sur l'existence et la non-existence de métriques Riemanniennes à courbure scalaire positive, invariantes par des actions propres; et on montre qu'une version localisée de la conjecture de Baum-Connes est plus faible que la conjecture originale, et on donne une description conceptuelle de la K-théorie des C * -algèbres de groupes.

We formulate, for any Lie group G acting isometrically on a manifold M , the general notion of a G-equivariant elliptic operator that is invertible outside of a G-cocompact subset of M . We prove a version of the Rellich lemma for this setting and use this to define the equivariant index of such operators. We show that Gequivariant Callias-type operators are self-adjoint, regular, and hence equivariantly invertible at infinity. Such operators explicitly arise from a pairing of the Dirac operator with the equivariant Higson corona. We apply the theory developed herein to obtain an obstruction to positive scalar curvature metrics on non-cocompact manifolds.

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For G an almost-connected Lie group, we study G-equivariant index theory for proper co-compact actions with various applications, including obstructions to and existence of G-invariant Riemannian metrics of positive scalar curvature. We prove a rigidity result for almost-complex manifolds, generalising Hattori's results, and an analogue of Petrie's conjecture. When G is an almost-connected Lie group or a discrete group, we establish Poincaré duality between G-equivariant K-homology and K-theory, observing that Poincaré duality does not necessarily hold for general G.