Local index formulae on noncommutative orbifolds and equivariant zeta functions for the affine metaplectic group

Abstract

We consider the algebra A of bounded operators on L2(Rn) generated by quantizations of isometric affine canonical transformations. The algebra A includes as subalgebras noncommutative tori of all dimensions and toric orbifolds. We define the spectral triple (A,H,D) with H=L2(Rn,Λ(Rn)) and the Euler operator D, a first order differential operator of index 1. We show that this spectral triple has simple dimension spectrum: For every operator B in the algebra Ψ(A,H,D) generated by the Shubin type pseudodifferential operators and the elements of A, the zeta function ζB(z)=Tr(B|D|−2z) has a meromorphic extension to C with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.