Transversally Elliptic Operators

In this paper we investigate the index theory of transversally elliptic pseudo-differential operators in the framework of noncommutative geometry. We give examples of spectral triples in the sense of Alain Connes and Henri Moscovici in [12] that are transversally elliptic but nonelliptic. We prove that these spectral triples satisfy the conditions which ensure the Connes-Moscovici local index formula applies. We show that such a spectral triple has discrete dimensional spectrum. We introduce an algebra Ψ⋊G consisting of families of of pseudodifferential operators, but with convolution like product over parameter space G. We show the symbolic calculus of this algebra, which is similar to Ψ∞, the algebra of pseudo-differential operators. Given a spectral triple on Ψ∞ ⋊G, we show that there is a finite number of trace-like functionals τ0, τ1, . . . , τN on Ψ ∞ ⋊ G which are defined and used in [12] for the computation of the local index formula. Those τ -functionals generalize the Wodzicki residue of Ψ∞ = Ψ∞ ⋊ {1}, which is τ0, the only nonzero among these τ functionals. Only the last nonzero τN is a trace on Ψ ∞⋊G. It is shown N is bounded by the sum of the dimensions of the compact Lie group G, and the underlying manifold M on which G acts. Moreover those τ -functionals, evaluated at A ∈ Ψ∞ ⋊G, are determined by the transversal symbol of A. The calculus on Ψ∞ ⋊G is helpful in the computation of Connes’ Chern character of a spectral triple. We show that Connes-Moscovici local index formula still works in the transversally elliptic case.