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Everything is a lagrangian submanifold (A. Weinstein's lagrangian creed...) Indeed, every manifold is the lagrangian zero section of its cotangent bundle ;)

But more serious: Weinstein's tubular neighbourhood theorem states that every lagrangian submanifolds submanifold in a symplectic manifold has a neighbourhood symplectomorphic to a neighbourhood of the zero section of the cotangent bundle.

Occurences of lagrangian submanifolds are indeed manifold: they arise as semiclassical support for certain FIO's and can also be thought of as semiclassical version of states in quantum mechanics a la via the WKB expansion. This point of view is exemplified a lot in the nice booklet of Bates and Weinstein.

Another occurence of coisotropics is in constraint mechanics: In Dirac's theory of constraints a coisotropic submanifold is what he calls a first class constraint. They arise very often in geometric mechanics with degenerate Lagrangians etc.

They are also the natural starting point for reduction: this is perhaps the more modern "coisotropic creed" of Lu (everything is a...)

Finally, to make the deformation quantization aspects a bit more precise: if you are looking for a submanifold $C \subseteq M$ in a Poisson manifold with star product $\star$ which allows for a (say) left module structure on $C^\infty(C)[[\hbar]]$ in such a way that the zeroth order of the module structure is the usual multiplication by the restriction, then you can show quite easily that $C$ has to be coisotropic. Martin Bordemann has a nice point of view how this relates to a theory of quantizing reduciton etc. in his (french!) big preprint :) In particular, the classical vanishing ideal becomes deformed into a left ideal for the star product (this was, I guess, essentially Lu's suggestion)

Note however, that there are other left ideals not of this form, e.g. the Gel'fand ideals of positive functionals, which can be much smaller.

The role of the lagrangian submanifolds $L$ in this context is that the corresponding representation of on $C^\infty(L)[[\hbar]]$ becomes "irreducible" in a meaningful way (trivial commutant in the local operators). However, this only makes sense in the symplectic surrounding. In a truely truly Poisson manifold, only coisotropic makes sense.

1

Everything is a lagrangian submanifold (A. Weinstein's lagrangian creed...) Indeed, every manifold is the lagrangian zero section of its cotangent bundle ;)

But more serious: Weinstein's tubular neighbourhood theorem states that every lagrangian submanifolds in a symplectic manifold has a neighbourhood symplectomorphic to a neighbourhood of the zero section of the cotangent bundle.

Occurences of lagrangian submanifolds are indeed manifold: they arise as semiclassical support for certain FIO's and can also be thought of semiclassical version of states in quantum mechanics a la WKB expansion. This point of view is exemplified a lot in the nice booklet of Bates and Weinstein.

Another occurence of coisotropics is in constraint mechanics: In Dirac's theory of constraints a coisotropic submanifold is what he calls a first class constraint. They arise very often in geometric mechanics with degenerate Lagrangians etc.

They are also the natural starting point for reduction: this is perhaps the more modern "coisotropic creed" of Lu (everything is a...)

Finally, to make the deformation quantization aspects a bit more precise: if you are looking for a submanifold $C \subseteq M$ in a Poisson manifold with star product $\star$ which allows for a (say) left module structure on $C^\infty(C)[[\hbar]]$ in such a way that the zeroth order of the module structure is the usual multiplication by the restriction, then you can show quite easily that $C$ has to be coisotropic. Martin Bordemann has a nice point of view how this relates to a theory of quantizing reduciton etc. in his (french!) big preprint :) In particular, the classical vanishing ideal becomes deformed into a left ideal for the star product (this was, I guess, essentially Lu's suggestion)

Note however, that there are other left ideals not of this form, e.g. the Gel'fand ideals of positive functionals, which can be much smaller.

The role of the lagrangian submanifolds $L$ in this context is that the corresponding representation of $C^\infty(L)[[\hbar]]$ becomes "irreducible" in a meaningful way (trivial commutant in the local operators). However, this only makes sense in the symplectic surrounding. In a truely Poisson manifold, only coisotropic makes sense.