On elliptic differential operators with shifts: II. The cohomological index formula

This paper is a continuation of [1], where we have studied a general class of (pseudo)differential operators with nonlocal coefficients, referred to as operators with shifts, and obtained a local index formula (i.e., a formula expressing the index as the integral of a differential form explicitly determined by the principal symbol of the operator) for matrix elliptic operators of this kind. In the present paper we finish the business by establishing a cohomological index formula of Atiyah-Singer type for elliptic differential operators with shifts acting between section spaces of arbitrary vector bundles. The key step is the construction of closed graded traces on certain differential algebras over the symbol algebra for this class of operators.

We do not formally assume the reader to be familiar with [1] as far as definitions are concerned but freely use the results obtained there. We also do not reproduce the discussion of general motivations for this research, which, as well as the bibliography, can be found in [1]. on M will be denoted by T , so that T g u = u(g -1 (x)),

x ∈ M.

We assume that Γ satisfies the following two conditions:

1. (Polynomial growth.) The group Γ is finitely generated, and the number of distinct elements of Γ representable by words of length ≤ k in some finite system of generators grows at most polynomially in k.

In what follows, we fix some system of generators and denote by |g| the minimum length of words representing g ∈ Γ.

2. (Diophantine property.) Let fix(g) be the set of fixed points of g ∈ Γ.

The estimate dist(g(x), x) ≥ C|g| -N dist(x, fix(g))

holds for some N, C > 0 and for all x ∈ M and g ∈ Γ. Here dist(x, fix(g)) is the Riemannian distance between x and the set fix(g), and by convention we set dist(x, fix(g)) = 1 if fix(g) is empty.

Matrix operators. Matrix pseudodifferential operators with shifts, ı.e., ΨDO with shifts acting on vector functions on M, can be described as follows.

(For more detail, see [1], where also further bibliographical references can be found.) A matrix ΨDO of order m with shifts has the form

where D g is a classical ΨDO of order m on M and the operators D g rapidly decay as |g| → ∞ in the natural Fréchet topology on the set of mth-order ΨDO.

Operators on sections of vector bundles. Pseudodifferential operators with shifts acting on sections of vector bundles are an easy generalization of matrix operators. To define them, one should localize into neighborhoods where the bundles are trivial. The only difference with the case of pseudodifferential operators without shifts is that our operators are no longer local, so we cannot localize into a neighborhood of the diagonal; hence two neighborhoods, instead of one, in the subsequent argument. Let E and F be finite-dimensional complex vector bundles on M. A linear operator

is called an mth-order ΨDO with shifts if for any trivializations of E and F over some neighborhoods U E , U F ⊂ M, respectively, and any functions ϕ ∈ C ∞ 0 (U E ) and ψ ∈ C ∞ 0 (U F ) the operator ψDϕ is an mth-order matrix ΨDO with shifts of the form (1).

We point out that no action of Γ on the bundles E and F is needed in this definition.

The linear space of mth-order pseudodifferential operators (2) with shifts will be denoted by Ψ m (E, F ) Γ . If E, F and H are three vector bundles on M, then the multiplication of operators induces a well-defined bilinear mapping

Just as for matrix operators, one readily proves that an mth-order ΨDO with shifts is a continuous operator of order m in the Sobolev spaces of sections of E and F .

Symbol: the matrix case. First, let us recall what happens in case the bundles E and F are trivial.

For the n × n ′ matrix operator (1), the symbol is defined by the formula

where the codifferential ∂g :

is the map induced by g (it acts as g along the base and as ((dg) * ) -1 in the fibers of S * M).

Symbol: the general case. If the operator ( 2) is a usual pseudodifferential operator, then its symbol is a bundle homomorphism π * E → π * F , where π : S * M → M is the natural projection. For pseudodifferential operators with shifts, which are highly nonlocal, this is no longer the case, and their symbols are defined as homomorphisms of section spaces of the bundles π * E and π * F rather than of the bundles themselves.

Definition 1. The symbol of the operator (2) is the operator

such that for any trivializations of E and F over some neighborhoods U E , U F ⊂ M, respectively, and any functions ϕ ∈ C ∞ 0 (U E ) and ψ ∈ C ∞ 0 (U F ) the operator ψσ(D)ϕ is the symbol of the operator ψDϕ.

One can readily verify that the symbol of a ΨDO with shifts is well defined. The space of symbols of ΨDO with shifts acting between section spaces of vector bundles E and F will be denoted by C ∞ (S * M, Hom(E, F )) Γ . For E = F , we use the notation C ∞ (S * M, End(E)) Γ , and for scalar symbols write C ∞ (S * M) Γ , just as in the first part of the paper. A generalization of the argument given