On an Extension of the Mean Index to a large subset of Linear Canonical Relations

Abstract

In this paper, viewing the symplectic linear group as a subset of the Lagrangian Grassmannian we extend the mean index to the complement of a codimension-two subset of the Grassmannian. This extension retains many of the desirable properties of the mean index, the most significant of which are continuity and a homogeneity condition adapted to the set-theoretic composition of canonical relations.