We define an equivariant index of Spin c -Dirac operators on possibly noncompact manifolds, acted on by compact, connected Lie groups. The main result in this paper is that the index decomposes into irreducible representations according to the quantisation commutes with reduction principle.

Equivariant indices, taking values in group-theoretic objects, have previously been defined in cases where either the group acting or the orbit space of the action is compact. In this paper, we define an equivariant index without assuming the group or the orbit space to be compact. This allows us to generalise an index of deformed Dirac operators, defined for compact groups by Braverman. In parts II and III of this series, we explore some properties and applications of this index.

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We prove a Fredholm property for spin-c Dirac operators D on non-compact manifolds satisfying a certain condition with respect to the action of a semi-direct product group K ⋉ Γ, with K compact and Γ discrete. We apply this result to an example coming from the theory of Hamiltonian loop group spaces. In this context we prove that a certain index pairing [X ] ∩ [D] yields an element of the formal completion R −∞ (T ) of the representation ring of a maximal torus T ⊂ H; the resulting element has an additional antisymmetry property under the action of the affine Weyl group, indicating [X ] ∩ [D] corresponds to an element of the ring of projective positive energy representations of the loop group.

In an earlier article we introduced a new definition for the 'quantization' of a Hamiltonian loop group space M, involving the equivariant L 2 -index of a Dirac-type operator D on a non-compact finite dimensional submanifold Y of M. In this article we study a Witten-type deformation of this operator, similar to the work of Tian-Zhang and Ma-Zhang. We obtain a formula for the index with infinitely many non-trivial contributions, indexed by the components of the critical set of the norm-square of the moment map. This is the main part of a new proof of the [Q, R] = 0 theorem for Hamiltonian loop group spaces.