# Local Index Formulae, Trace Expansions, and Noncommutative Residues on an Algebra of Fourier Integral Operators

We study two subalgebras $\mathscr A$ and $\mathscr B$ of $\mathscr L(L^2(\mathbb R^n))$. $\mathscr A$ is generated by quantizations of affine isometric canonical transformations; its elements are finite sums

\[\sum_{g\in U(n), w\in \mathbb C^n} a_{g,z}R_g T_{w},\]

where $a_{g,z}\in \mathbb C$, $T_{w}$, $w=a-ik\in \mathbb C^n\cong \mathbb R^{2n}$, is a Heisenberg-Weyl operator given by $T_{w} u(x) =e^{ikx-iak/2}u(x-a)$, and $R_{g}$ is a metaplectic operator associated to the unitary $g$ in $\mathbb C^n$.

As particular cases, this algebra encompasses noncommutative tori of arbitrary dimension and noncommutative toric orbifolds.

We then introduce a Dirac type operator $D$ for which $(\mathscr A, L^2(\mathbb R^n), D)$ is a spectral triple in the sense of Connes and compute explicitly the cyclic cocycles in the Connes-Moscovici local index formula.

Next, we consider the larger algebra $\mathscr B$ consisting of all bounded operators on $L^2(\mathbb R^n)$ that can be written as finite sums

\[ B= \sum R_g T_w A,\]

where $R_g$ and $T_w$ are as above and $A$ is a Shubin type pseudodifferential operator. Choosing a reference operator such as $H=|x|^2-\Delta$ (other choices are possible), we obtain

-- expansions for the trace of $B(H-\lambda)^{-K}$ in powers of $\lambda$ and $\log \lambda$ for $K$ sufficiently large, as $\lambda \to \infty$ in a sector of $\mathbb C$

-- expansions for the trace of $Be^{-tH}$ as $t\to 0^+$ in powers of $t$ and $\log t$,

-- the meromorphic structure of the zeta function $\zeta_B(z) = \text{Tr} (BH^{-z})$, defined initially for large $\text{Re}(z)$ and then extended to $\mathbb C$.

These three statements are essentially equivalent via suitable transition formulae.

Moreover, we construct a noncommutative residue that extends the Wodzicki residue to this situation and define equivariant traces on $\mathscr B$.

(Joint work with Anton Savin, RUDN University, Moscow)