An extension of the Duistermaat-Singer Theorem to the semi-classical Weyl algebra

TL;DR

This paper extends the Duistermaat-Singer Theorem to the semi-classical Weyl algebra, showing that automorphisms are given by conjugation with elliptic Fourier Integral Operators, with linear complex symplectic maps replacing nonlinear symplectic diffeomorphisms.

Contribution

It generalizes the classical theorem to the semi-classical setting, characterizing automorphisms of the semi-classical Weyl algebra as conjugations by elliptic Fourier Integral Operators.

Findings

Automorphisms are given by conjugation with elliptic Fourier Integral Operators.

Linear complex symplectic maps replace nonlinear symplectic diffeomorphisms.

The structure is localized at a single point.

Abstract

Motivated by many recent works (by L. Charles, V. Guillemin, T. Paul, J. Sj\"ostrand, A. Uribe, S. Vu Ngoc, S. Zelditch and others) on the semi-classical Birkhoff normal forms, we investigate the structure of the group of automorphisms of the graded semi-classical Weyl algebra which is used to get the normal forms. The answer is quite similar to the Theorem of Duistermaat and Singer for the usual algebra of pseudo-differential operators where all automorphisms are given by conjugation by an elliptic Fourier Integral Operator (a FIO). Here what replaces general non-linear symplectic diffeomeorhisms is just linear complex symplectic maps, because everything is localized at a single point.