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We use the properties of $Mp\left(n\right)$ to construct functions ${\mu }_{\ell }:Mp\left(n\right)\to {\mathbf{Z}}_8$ associated with the elements $\ell$ of the lagrangian grassmannian $\Lambda$ (n) which generalize the Maslov index on Mp(n) defined by J. Leray in his “Lagrangian Analysis”. We deduce from these constructions the identity between $Mp\left(n\right)$ and a subset of $Sp\left(n\right)\times {\mathbf{Z}}_8$, equipped with appropriate algebraic and topological structures.

This work is devoted to a systematic study of the microlocal regularity properties of pseudo-differential operators with the transmission property. We introduce a “boundary singular spectrum”, denoted $\partial W{F}_{\omega }\left(u\right)$ for distributions $u\in D^{\text{'}}({\mathbf{R}}_{+}^n)$, regular in the normal variable $x_n$ (thus, $\partial W{F}_{\omega }\left(u\right)=\varnothing$ means that $u\in {\cap }_{s+t=1/2}{H}^{s+t}$ near the boundary), and it is shown that $\partial W{F}_{\omega -m}\left[P\right(u^0{)}_{x_n\>0}]$ is a subset of $\partial WF\left(u\right)$ if $P$ has degree $m$ and the transmission property. We finally prove that these results can bef used to examinate the (microlocal) regularity of the solutions of differential...