A note on traces for the Heisenberg calculus
Alexander Gorokhovsky, Erik van Erp
TL;DR
This work clarifies the trace on the symbol algebra ${\mathscr S}_H$ of the Heisenberg calculus by tying it to traces defined on general homogeneous groups and to central evaluations in the Heisenberg group. The central result expresses the trace ${\tau}$ as a precise linear combination of central traces ${\tau_+}$ and ${\tau_-}$ via ${\tau} = \frac{(2\pi)^{n+1}}{2(n!) i^{n+1}}(\tau_+ - (-1)^n\tau_-)$, enabling a simpler formulation of index calculations. The approach uses Fourier-analytic connections to the Weyl algebra and introduces a residue trace ${\mathrm{Res}}(k)$ to decompose ${\tau}_z$; the framework links back to the Connes-Chern character pairing from earlier work, and clarifies the trace structure underlying the Heisenberg index formula. A technical Fourier-analytic lemma about limits of Fourier transforms of homogeneous distributions under regularization underpins the main decomposition and its rigorous justification.
Abstract
In previous work, we gave a local formula for the index of Heisenberg elliptic operators on contact manifolds. We constructed a cocycle in periodic cyclic cohomology which, when paired with the Connes-Chern character of the principal Heisenberg symbol, calculates the index. A crucial ingredient of our index formula was a new trace on the algebra of Heisenberg pseudodifferential operators. The construction of this trace was rather involved. In the present paper, we clarify the nature of this trace.