The Maslov cycle as a Legendre singularity and projection of a wavefront set

TL;DR

This paper explores the geometric and analytical structure of the Maslov cycle, demonstrating its interpretation as a Legendre singularity and linking it to wavefront sets of Fourier integral distributions.

Contribution

It establishes that the Maslov cycle is a Legendre singularity and identifies the associated Lagrangian submanifold as the wavefront set of a specific Fourier integral distribution.

Findings

Maslov cycle is a Legendre singularity.

The Lagrangian submanifold S is the wavefront set of a Fourier integral distribution.

S is obtained as the evaluation at 0 of the quantizations.

Abstract

A Maslov cycle is a singular variety in the lagrangian grassmannian L(V) of a symplectic vector space V consisting of all lagrangian subspaces having nonzero intersection with a fixed one. Givental has shown that a Maslov cycle is a Legendre singularity, i.e. the projection of a smooth conic lagrangian submanifold S in the cotangent bundle of L(V). We show here that S is the wavefront set of a Fourier integral distribution which is "evaluation at 0 of the quantizations".