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 $L^2({\mathbb{R}}^n)$ generated by quantizations of isometric affine canonical transformations. The algebra $A$ includes as subalgebras all noncommutative tori and toric orbifolds. We define the spectral triple $(A,H,D)$ with $H=L^2({\mathbb{R}}^n,\Lambda ({\mathbb{R}}^n))$ 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 $\Psi (A,H,D)$ generated by the Shubin type pseudodifferential operators and the elements of $A$, the zeta function ${\zeta }_B(z)=\mathrm{Tr}(B|D{|}^{-2z})$ has a meromorphic extension to $\mathbb{C}$ with at most simple poles. Our main result then is an explicit algebraic expression for the Connes-Moscovici cyclic cocycle. As a corollary we obtain local index formulae for noncommutative tori and toric orbifolds.

Accepted for publication in Adv. Math