Symplectic Carleman approximation on co-adjoint orbits
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Wold, Erlend Fornæss
For a complex Lie group G with a real form G0⊂G, we prove that any Hamiltionian automorphism ϕ of a coadjoint orbit O0 of G0 whose connected components are simply connected, may be approximated by holomorphic O0-invariant symplectic automorphism of the corresponding coadjoint orbit of G in the sense of Carleman, provided that O is closed. In the course of the proof, we establish the Hamiltonian density property for closed coadjoint orbits of all complex Lie groups.
 Thanks to the organizers for inviting me. Yes, so I'm going to talk about Simplectic Carleman approximation on joint orbits. The starting point is a question asked by Nassim, that if you take a smooth, C infinity smooth, Simplectomorphism on R2N, R2N sits inside C2N, then the question is if this is a proximal in the sense of Carleman by elements of this holomorphic Simplectic group leaving R2N invariant. And of course Simplectic on R2 just means that phi preserves this Simplectic form and then the analog thing for the holomorphic in the holomorphic case. And then what does this Carleman approximation mean? This means that for any continuous function on R2N, strictly positive, there should exist a holomorphic real Simplectomorphism of C2N so that this psi approximates phi to precision epsilon of x for x on R2N. So now the point is that this is much more than uniform approximation or even approximation on compacts so this epsilon could go to zero as fast as you want as x goes to infinity. And if you can also, we will do that later, throw in some derivatives here you want to approximate. So anyhow I thought I'd just start by, before I start discussing this, just go through a little bit of the history of Carleman approximation for functions in case many of you didn't see that before. So a little history, so this is not Carleman approximation but something maybe all of you know that Weierstrass proved that any continuous function on an interval AB is uniformly approximable by polynomials. So the Junger theorem tells you if K is a compact set with C minus K connected then any function F holomorphic on K, so this means holomorphic on a neighborhood of K, maybe approximated by polynomials. And then comes this Carleman theorem which is a generalization of Weierstrass telling that you can replace the closed interval by the real line. And then there is the corresponding statement that I had before, any continuous function on the entire real line can be approximated to precision epsilon of x where x is a continuous function. So maybe such a theorem is a bit surprising if you did not see it before but if you have a slight generalization of the two theorems above it's very easy to prove. So I thought I'll just give a short sketch for later. So well you need a combination theorem, a Marguer-Lian type theorem but you can just imagine if you have a ball, maybe say a disc of radius r in the complex plane and you consider the real line r and then I'm going to start with a continuous function F, it's holomorphic on this disc and continuous on the real line. So if you assume that you have a combination of those two theorems above, so just assume, see that there exists a sequence of polynomials approximating, converging to F on this closed disc union and let's say the closed disc of radius r plus 2, disc intersected r. So I assume that I can approximate this given function on that disc, union such an interval then you can just, you take a cutoff function which is 0 on the neighborhood of the disc of radius r plus 1 then it dies when you come out of the disc of radius r plus 2, you choose such a cutoff function, then you just write the Hj is Gj plus cutoff function, what do you want F minus Gj and this function is going to converge to F uniformly on the real line and it approximates the original F on this disc and it's holomorphic on a disc of radius r plus 1. So now you can imagine you just put this into an inductive procedure and you get this approximation result here. So this was just to emphasize that when we deal with this kind of thing, there are two things. So one thing is you have to prove a local theorem compact result and then you need some inductive procedure to go to a limit. Okay, anyhow, so just keep that in mind for later. This was Kalman, more generally it's proved by these guys that, so this is a complete characterization that the closed subset X in a complex plane satisfies Kalman approximation if and only if it has no interior and the complement is connected and locally connected at infinity. Alright, so that's one complex variable. A little bit closer to our case, Rn inside Cn satisfies Kalman approximation and the proof is just not very much harder than this. It's quite simple. Moving away from flat things, it's known that the smooth unbounded curve always admits such approximation and a bit more generally these local rectifiable curves. However, moving up to higher dimensional submanifolds, things now change in several complex variables because there are polynomially convex totally real submanifolds which do not admit Kalman approximation. So these, of course, admits approximation on compact sets can be exhausted by polynomially convex set but there is some global business going on that prevents this Kalman approximation to hold. And of course what, well, maybe not of course, but if you go back to this Kelditsch-Lavrentief characterization you see that they require, it's necessary that the complement is locally connected at infinity and somehow this is, well, that's not correct to say, but that's what goes wrong in this case except that locally connected at infinity doesn't make sense in this setting so you have to reformulate and if you reformulate what that means you can characterize completely totally real submanifolds of CN which admit Kalman approximation. So they have to be polynomially convex and I'm not going to explain this, have something called bounded exhaustion holes. But this is certainly something that holds for Rn in CN and in particular R2n into in C2n. All right, moving to holomorphic maps, it's known that a proper smooth embedding from R to C2 can be approximated in the sense of Kalman by proper holomorphic embeddings of the complex plane itself and it is known and this is a little bit closer to what we are going to discuss today that if you are in CN and you look at, so I'm not going to be too precise, so I choose K strictly less than N and I'm going to assume that I have a totally real manifold that's sort of, let's say just tangentially equal to this so it doesn't matter what it looks like down here but it should look like Rk near infinity then if you choose, so this is a bit maybe unfortunate language if you choose a smooth embedding of this M which is the identity on M outside a compact set then you can approximate it in this Kalman sense by holomorphic automorphisms of CN. All right and then finally, finally the result today, so this is joint work with Fuxiang Deng precisely it's a positive answer to the question that we started with you take a simplectomorphism of R2N you let epsilon of X be a strictly positive function continuous and fix an integer K then you can Kalman approximate by holomorphic simplectomorphisms also derivatives up to order K. All right so this is very different so and the requirement that R2N is left invariant right so this is a little bit different from this situation here so here we assume it's necessary to assume that K is less than N here it's not it's exactly the same number right so here 2N is the K so the totally real guy here has maximal dimension and you want to leave it invariant this is not what we want to do here. All right so this is the main theorem but this is sort of a special case over more general result so you can produce many more like totally real submanifolds of complex symplectic manifolds for which such theorems hold so I'm going to start by just describing a just more general setup. Sorry. Yes, yes, yes, yes, I don't think you can. This is the order of derivative. I think if you specify K first you could say that phi has to be CK plus something you are going to lose some derivatives. All right so in the title I had coadjoint orbits so I'm going to say something about that so I'm going to start with a Lie group and I look at the tangent space of the group then if you fix an element of the Lie group you have a conjugation map so this is just standard things this conjugation map fixes the identity right which means that it's differential acts on the tangent space at identity right and that differential is called the adjoint action of G on the group. All right so you want coadjoint action so now you want an action on the dual space on the cotangent space and then you just define by duality so you said that add star that's the coadjoint action at the point G no applied to G acting on a cotangent is just defined by take the adjoint action of the inverse and let the cotangent act on that. All right so then you get an action on the dual space on the cotangent space. And as soon as you fix so you fix such a such a psi and then you take like G act on your the whole group G act on your cotangent space you get an orbit so that's a smooth sub manifold of the cotangent space not necessarily closed but I'm going to assume I'm only going to consider closed ones in this talk. And what does this have to do with then the question if we look at an example you look at the Lie group so let's say a real Lie group consisting of matrices like this this you identify with R3 tangent space you identify with R3 and if you just compute what this coadjoint action is it's given by this. So for a and b this is the coadjoint action and you see that C is gone there is no C and then you also see that if you fix X3 then this orbit is just going to be an R2 if you vary b and a in R2 so you just get the flat R2 sitting inside of sitting inside of R3. So that's the coadjoint orbit R2 and if you just complexify this group you just throw in complex coefficients instead then you get the coadjoint orbit which is just the complexification of that R2 so R2 just sits inside C2 and that's the special case that we considered before R2 well in the N is 1 that R2 sits inside C2 and we do want to do approximation. But at this point we did not introduce there is no symplectic structure here yet so what do you do so let's see how do we get some vector fields on I want to define some vector fields on my coadjoint orbit and then I want to define a symplectic form afterwards so if you take a V in the tangent space you can consider the exponent map e to the Tv so you can take the adjoint coadjoint action of the exponential map you can apply it to a point in the dual space right. So then this one parameter family for T0 you start on your point xi and then you just flow inside your coadjoint orbit and so if you differentiate you get a vector field right so I just take d dt oh now it jumped at T equals 0 you get you get you get a vector field and it's complete you can note for later it's defined by differentiating a flow right so this is a complete smooth vector field if we look at real coadjoint orbit or holomorphic if we look at the complex one. All right so and this is also important for later that now we have lots of complete whole smooth or holomorphic vector fields on our coadjoint orbits by differentiating these flows all right and yeah this is a complete whole vector field then for two vector fields you want to define the symplectic form you just at the point xi then you take the Lie bracket of the two generators for the vector field so that's a new tangent vector and then the coad tangent can act on this bracket okay and then of course there is something to prove that this is a symplectic form but it is a symplectic form I'm not going to prove that but that's that's a standard standard thing and it's the fact that this vector field that we have above is Hamiltonian okay so it satisfies this contract xv with omega it's cost okay so this is symplectic vector field. All right but now comes the important thing for us is that you can also I mean if you want to define a symplectic or a Hamiltonian vector field you can do that by using potentials yeah you take a smooth function f on your whatever space symplectic manifold and if you find you can find at least no you can find the vector field x just satisfies df is the contraction of omega with x then x is a symplectic vector field with potential f okay and now comes the important thing here that this I think I have it on the next slide that I can say so this complete vector field has a potential and the potential is v okay so v is something in the tangent space it can act on the cotangent space so it's a function it could be a or you can use it as a potential and what happens is that you just get that vector field back okay so this is the important thing that all that all linear maps on your coad joint orbit are potentials for complete vector fields okay so this is the important thing this is the important thing so now yes so now I'm just going to forget a little bit this so what we are going to do we are going to look at this real coad joint orbits similar to the Heisenberg group sitting inside of complex coad joint orbits as closed sub manifolds and we have this property here so I'm just going to forget the orbits and just formulate just a general framework which is quite simple so I'm just going to say okay let's say we are in CN is RN plus IRN I'm just going to assume I have a closed sub manifold and I'm going to intersect it with RN and get a smooth closed manifold with the corresponding real dimension equal to the maximal real dimension inside Z I'm going to assume I have a symplectic form on my complex manifold so that when I restrict to the real guy I get the real symplectic form and I'm going to assume that whenever I take a linear map on CN and I use it as a potential for a Hamiltonian vector field on Z then that vector field is complete okay so those were the crucial facts for the coad joint orbits that we had before and this is certainly the case for R2N sitting inside of C2N okay all right all right so let's see this is going a bit faster than I thought let's try to look at this approximation of symplectic, symplectomorphism on the real center here and then as I said here so there are two things you have to do you need to prove some local approximation statement something on compacts and then you need some inductive scheme to build a global object as an inductive limit okay and somehow this induction scheme here gets a little bit complicated it's a little bit messy so I'm just going to stick I'm gonna just explain the compact the compact approximation and then well maybe we can indicate indicate later okay so what do we do so I'm going to consider ah so now I have to make some assumption so if if so if I am on R2N and I have a symplectomorphism phi and if it fixes the origin I can put phi T of x is 1 over T phi of Tx so this is an isotope an isotope of symplectomorphisms so that when T is 0 you have the identity map and when T is 1 you have your original phi okay so any symplectic diffeomorphism on R2N is in the C infinity path connected component of the identity okay so this is packed on R2N and if I work in this this abstract setting I just have to assume that I start with a symplectomorphism which is in the path connected component of the identity so that just has to be an assumption so then we take this this phi T so then you can interpret phi T as the flow of this time dependent vector field you just differentiate with respect to T right so now your goal is to approximate approximate the flow of a time dependent vector field so now it lives on your lives on your real coadjoint orbit assume or real z0 then it's a standard thing that if you just divide your time interval into many many many many small intervals and you just fix the time at the end point so these small intervals and you consider the flows of the time independent vector fields you can approximate the flow of this by just composing by composing flows of time independent vector so that's a standard thing so now you can forget about the time and ask to approximate the time independent vector field here alright so now x is time independent Hamiltonian so now I have to make another assumption and that is that and that is that my z0 is simply connected because then I can find the potential so the smooth vector field x has a potential f come that potential can be approximated by real holomorphic polynomials p of z so just holomorphic polynomials with real coefficients since this is e0 6 inside our end okay so now the points now I have now I have a real holomorphic polynomial p of z and since it's real if I use the new the polynomial as a potential for a new for a new vector field it is going to be tangent to the real z0 since it's real so I'm going to get the holomorphic vector field on the big complex manifold but it's going to be tangent along this real guy real center alright so now I have a polynomial now it's a standard thing that any polynomial in a number of variables can be written as a one variable polynomial in linear forms on well Rn or Cn okay so lambda jz is just a linear map linear linear map from the CN okay so this you can do now you are going to use the assumption so I assumed that if you if you forget the power mj and just use the linear guy lambda xj as a potential then you get something complete that was that was the assumption which we had automatically for this coad joint orbits but this means that this lambda j if you lose use the lambda j to the mj is still complete okay because the lambda if you look at the vector fields you get from the lambda j this lambda j to the mj is going to be constant on the level on the flow curves so this is still going to be complete so now if you use p I mean you use each each component here of p as a potential right individually what you did was you wrote wrote x as a sum of complete holomorphic vector fields and they come from real polynomials so they are tangent to to z0 and then you are done approximating on a compact set because it's a standard thing that if you have a sum of holomorphic no smooth or holomorphic for that matter if you have a sum of vector fields and you want to approximate the flow then you can flow along the individual vector fields for small amounts of time and compose like this here so when n goes to infinity you are approximating Phi t and each of the individual each of the individual components here are simplectic diffeomorphisms because they are time t over n flows of complete vector fields alright so this means yes I'm not really gonna so this means so you start with something z not it sits inside some larger complex manifold z and you have to choose so a compact we have the diffeomorphism Phi choose compact set k 1 and what we achieved was approximating this Phi on this particular compact case but this Phi is going to do something arbitrary outside here right so this means that let's see so now it had maybe a psi 1 or something so this is not really good you have to correct it so you have to do something like considering okay now you rather want psi 1 inverse composed with Phi because if you compose this with this you're back in business so now you try to approximate this and then correct your original approximation and then you try to set up inductive scheme to exhaust all of the not but of course you have to take care that your sequence of maps also converges to a diffeomorphism of C to N and that makes it a little bit tricky but that's basically the idea and I think actually that I am done so thank you.
 Thanks to the organizers for inviting me.  Yes, so I'm going to talk about Simplectic Carleman approximation on joint orbits.  The starting point is a question asked by Nassim, that if you take a smooth, C infinity  smooth, Simplectomorphism on R2N, R2N sits inside C2N, then the question is if this is  a proximal in the sense of Carleman by elements of this holomorphic Simplectic group leaving  R2N invariant.  And of course Simplectic on R2 just means that phi preserves this Simplectic form and  then the analog thing for the holomorphic in the holomorphic case.  And then what does this Carleman approximation mean?  This means that for any continuous function on R2N, strictly positive, there should exist  a holomorphic real Simplectomorphism of C2N so that this psi approximates phi to precision  epsilon of x for x on R2N.  So now the point is that this is much more than uniform approximation or even approximation  on compacts so this epsilon could go to zero as fast as you want as x goes to infinity.  And if you can also, we will do that later, throw in some derivatives here you want to  approximate.  So anyhow I thought I'd just start by, before I start discussing this, just go through a  little bit of the history of Carleman approximation for functions in case many of you didn't see  that before.  So a little history, so this is not Carleman approximation but something maybe all of you  know that Weierstrass proved that any continuous function on an interval AB is uniformly approximable  by polynomials.  So the Junger theorem tells you if K is a compact set with C minus K connected then any  function F holomorphic on K, so this means holomorphic on a neighborhood of K, maybe  approximated by polynomials.  And then comes this Carleman theorem which is a generalization of Weierstrass telling  that you can replace the closed interval by the real line.  And then there is the corresponding statement that I had before, any continuous function  on the entire real line can be approximated to precision epsilon of x where x is a continuous  function.  So maybe such a theorem is a bit surprising if you did not see it before but if you have  a slight generalization of the two theorems above it's very easy to prove.  So I thought I'll just give a short sketch for later.  So well you need a combination theorem, a Marguer-Lian type theorem but you can just  imagine if you have a ball, maybe say a disc of radius r in the complex plane and you consider  the real line r and then I'm going to start with a continuous function F, it's holomorphic  on this disc and continuous on the real line.  So if you assume that you have a combination of those two theorems above, so just assume,  see that there exists a sequence of polynomials approximating, converging to F on this closed  disc union and let's say the closed disc of radius r plus 2, disc intersected r.  So I assume that I can approximate this given function on that disc, union such an interval  then you can just, you take a cutoff function which is 0 on the neighborhood of the disc  of radius r plus 1 then it dies when you come out of the disc of radius r plus 2, you choose  such a cutoff function, then you just write the Hj is Gj plus cutoff function, what do  you want F minus Gj and this function is going to converge to F uniformly on the real line  and it approximates the original F on this disc and it's holomorphic on a disc of radius  r plus 1.  So now you can imagine you just put this into an inductive procedure and you get this approximation  result here.  So this was just to emphasize that when we deal with this kind of thing, there are two  things.  So one thing is you have to prove a local theorem compact result and then you need some  inductive procedure to go to a limit.  Okay, anyhow, so just keep that in mind for later.  This was Kalman, more generally it's proved by these guys that, so this is a complete  characterization that the closed subset X in a complex plane satisfies Kalman approximation  if and only if it has no interior and the complement is connected and locally connected  at infinity.  Alright, so that's one complex variable.  A little bit closer to our case, Rn inside Cn satisfies Kalman approximation and the  proof is just not very much harder than this.  It's quite simple.  Moving away from flat things, it's known that the smooth unbounded curve always admits  such approximation and a bit more generally these local rectifiable curves.  However, moving up to higher dimensional submanifolds, things now change in several complex variables  because there are polynomially convex totally real submanifolds which do not admit Kalman  approximation.  So these, of course, admits approximation on compact sets can be exhausted by polynomially  convex set but there is some global business going on that prevents this Kalman approximation  to hold.  And of course what, well, maybe not of course, but if you go back to this Kelditsch-Lavrentief  characterization you see that they require, it's necessary that the complement is locally  connected at infinity and somehow this is, well, that's not correct to say, but that's  what goes wrong in this case except that locally connected at infinity doesn't make sense in  this setting so you have to reformulate and if you reformulate what that means you can  characterize completely totally real submanifolds of CN which admit Kalman approximation.  So they have to be polynomially convex and I'm not going to explain this, have something  called bounded exhaustion holes.  But this is certainly something that holds for Rn in CN and in particular R2n into in  C2n.  All right, moving to holomorphic maps, it's known that a proper smooth embedding from  R to C2 can be approximated in the sense of Kalman by proper holomorphic embeddings of  the complex plane itself and it is known and this is a little bit closer to what we are  going to discuss today that if you are in CN and you look at, so I'm not going to be  too precise, so I choose K strictly less than N and I'm going to assume that I have a totally  real manifold that's sort of, let's say just tangentially equal to this so it doesn't matter  what it looks like down here but it should look like Rk near infinity then if you choose,  so this is a bit maybe unfortunate language if you choose a smooth embedding of this M  which is the identity on M outside a compact set then you can approximate it in this Kalman  sense by holomorphic automorphisms of CN.  All right and then finally, finally the result today, so this is joint work with Fuxiang  Deng precisely it's a positive answer to the question that we started with you take a simplectomorphism  of R2N you let epsilon of X be a strictly positive function continuous and fix an integer  K then you can Kalman approximate by holomorphic simplectomorphisms also derivatives up to  order K.  All right so this is very different so and the requirement that R2N is left invariant  right so this is a little bit different from this situation here so here we assume it's  necessary to assume that K is less than N here it's not it's exactly the same number right  so here 2N is the K so the totally real guy here has maximal dimension and you want to  leave it invariant this is not what we want to do here.  All right so this is the main theorem but this is sort of a special case over more  general result so you can produce many more like totally real submanifolds of complex  symplectic manifolds for which such theorems hold so I'm going to start by just describing  a just more general setup.  Sorry.  Yes, yes, yes, yes, I don't think you can.  This is the order of derivative.  I think if you specify K first you could say that phi has to be CK plus something you are  going to lose some derivatives.  All right so in the title I had coadjoint orbits so I'm going to say something about  that so I'm going to start with a Lie group and I look at the tangent space of the group  then if you fix an element of the Lie group you have a conjugation map so this is just  standard things this conjugation map fixes the identity right which means that it's differential  acts on the tangent space at identity right and that differential is called the adjoint  action of G on the group.  All right so you want coadjoint action so now you want an action on the dual space on  the cotangent space and then you just define by duality so you said that add star that's  the coadjoint action at the point G no applied to G acting on a cotangent is just defined  by take the adjoint action of the inverse and let the cotangent act on that.  All right so then you get an action on the dual space on the cotangent space.  And as soon as you fix so you fix such a such a psi and then you take like G act on your  the whole group G act on your cotangent space you get an orbit so that's a smooth sub manifold  of the cotangent space not necessarily closed but I'm going to assume I'm only going to  consider closed ones in this talk.  And what does this have to do with then the question if we look at an example you look  at the Lie group so let's say a real Lie group consisting of matrices like this this you  identify with R3 tangent space you identify with R3 and if you just compute what this  coadjoint action is it's given by this.  So for a and b this is the coadjoint action and you see that C is gone there is no C and  then you also see that if you fix X3 then this orbit is just going to be an R2 if you  vary b and a in R2 so you just get the flat R2 sitting inside of sitting inside of R3.  So that's the coadjoint orbit R2 and if you just complexify this group you just throw  in complex coefficients instead then you get the coadjoint orbit which is just the  complexification of that R2 so R2 just sits inside C2 and that's the special case that  we considered before R2 well in the N is 1 that R2 sits inside C2 and we do want to  do approximation.  But at this point we did not introduce there is no symplectic structure here yet so what  do you do so let's see how do we get some vector fields on I want to define some vector  fields on my coadjoint orbit and then I want to define a symplectic form afterwards so  if you take a V in the tangent space you can consider the exponent map e to the Tv so  you can take the adjoint coadjoint action of the exponential map you can apply it to  a point in the dual space right.  So then this one parameter family for T0 you start on your point xi and then you just flow  inside your coadjoint orbit and so if you differentiate you get a vector field right  so I just take d dt oh now it jumped at T equals 0 you get you get you get a vector field  and it's complete you can note for later it's defined by differentiating a flow right so  this is a complete smooth vector field if we look at real coadjoint orbit or holomorphic  if we look at the complex one.  All right so and this is also important for later that now we have lots of complete whole  smooth or holomorphic vector fields on our coadjoint orbits by differentiating these  flows all right and yeah this is a complete whole vector field then for two vector fields  you want to define the symplectic form you just at the point xi then you take the Lie  bracket of the two generators for the vector field so that's a new tangent vector and then  the coad tangent can act on this bracket okay and then of course there is something to prove  that this is a symplectic form but it is a symplectic form I'm not going to prove that  but that's that's a standard standard thing and it's the fact that this vector field that  we have above is Hamiltonian okay so it satisfies this contract xv with omega it's cost okay  so this is symplectic vector field.  All right but now comes the important thing for us is that you can also I mean if you  want to define a symplectic or a Hamiltonian vector field you can do that by using potentials  yeah you take a smooth function f on your whatever space symplectic manifold and if  you find you can find at least no you can find the vector field x just satisfies df is  the contraction of omega with x then x is a symplectic vector field with potential f  okay and now comes the important thing here that this I think I have it on the next slide  that I can say so this complete vector field has a potential and the potential is v okay  so v is something in the tangent space it can act on the cotangent space so it's a function  it could be a or you can use it as a potential and what happens is that you just get that  vector field back okay so this is the important thing that all that all linear maps on your  coad joint orbit are potentials for complete vector fields okay so this is the important  thing this is the important thing so now yes so now I'm just going to forget a little  bit this so what we are going to do we are going to look at this real coad joint orbits  similar to the Heisenberg group sitting inside of complex coad joint orbits as closed sub  manifolds and we have this property here so I'm just going to forget the orbits and just  formulate just a general framework which is quite simple so I'm just going to say okay  let's say we are in CN is RN plus IRN I'm just going to assume I have a closed sub manifold  and I'm going to intersect it with RN and get a smooth closed manifold with the corresponding  real dimension equal to the maximal real dimension inside Z I'm going to assume I have a symplectic  form on my complex manifold so that when I restrict to the real guy I get the real symplectic  form and I'm going to assume that whenever I take a linear map on CN and I use it as  a potential for a Hamiltonian vector field on Z then that vector field is complete okay  so those were the crucial facts for the coad joint orbits that we had before and this is  certainly the case for R2N sitting inside of C2N okay all right all right so let's see  this is going a bit faster than I thought let's try to look at this approximation of  symplectic, symplectomorphism on the real center here and then as I said here so there  are two things you have to do you need to prove some local approximation statement something  on compacts and then you need some inductive scheme to build a global object as an inductive  limit okay and somehow this induction scheme here gets a little bit complicated it's a  little bit messy so I'm just going to stick I'm gonna just explain the compact the compact  approximation and then well maybe we can indicate indicate later okay so what do we do so I'm  going to consider ah so now I have to make some assumption so if if so if I am on R2N  and I have a symplectomorphism phi and if it fixes the origin I can put phi T of x is  1 over T phi of Tx so this is an isotope an isotope of symplectomorphisms so that when  T is 0 you have the identity map and when T is 1 you have your original phi okay so any  symplectic diffeomorphism on R2N is in the C infinity path connected component of the  identity okay so this is packed on R2N and if I work in this this abstract setting I  just have to assume that I start with a symplectomorphism which is in the path connected component  of the identity so that just has to be an assumption so then we take this this phi T  so then you can interpret phi T as the flow of this time dependent vector field you just  differentiate with respect to T right so now your goal is to approximate approximate the  flow of a time dependent vector field so now it lives on your lives on your real coadjoint  orbit assume or real z0 then it's a standard thing that if you just divide your time interval  into many many many many small intervals and you just fix the time at the end point so  these small intervals and you consider the flows of the time independent vector fields  you can approximate the flow of this by just composing by composing flows of time independent  vector so that's a standard thing so now you can forget about the time and ask to approximate  the time independent vector field here alright so now x is time independent Hamiltonian so  now I have to make another assumption and that is that and that is that my z0 is simply connected  because then I can find the potential so the smooth vector field x has a potential f come  that potential can be approximated by real holomorphic polynomials p of z so just holomorphic  polynomials with real coefficients since this is e0 6 inside our end okay so now the points  now I have now I have a real holomorphic polynomial p of z and since it's real if I use the new  the polynomial as a potential for a new for a new vector field it is going to be tangent  to the real z0 since it's real so I'm going to get the holomorphic vector field on the  big complex manifold but it's going to be tangent along this real guy real center alright  so now I have a polynomial now it's a standard thing that any polynomial in a number of  variables can be written as a one variable polynomial in linear forms on well Rn or Cn  okay so lambda jz is just a linear map linear linear map from the CN okay so this you can  do now you are going to use the assumption so I assumed that if you if you forget the  power mj and just use the linear guy lambda xj as a potential then you get something  complete that was that was the assumption which we had automatically for this coad joint  orbits but this means that this lambda j if you lose use the lambda j to the mj is still  complete okay because the lambda if you look at the vector fields you get from the lambda  j this lambda j to the mj is going to be constant on the level on the flow curves so this is  still going to be complete so now if you use p I mean you use each each component here  of p as a potential right individually what you did was you wrote wrote x as a sum of  complete holomorphic vector fields and they come from real polynomials so they are tangent  to to z0 and then you are done approximating on a compact set because it's a standard thing  that if you have a sum of holomorphic no smooth or holomorphic for that matter if you have  a sum of vector fields and you want to approximate the flow then you can flow along the individual  vector fields for small amounts of time and compose like this here so when n goes to infinity  you are approximating Phi t and each of the individual each of the individual components  here are simplectic diffeomorphisms because they are time t over n flows of complete vector  fields alright so this means yes I'm not really gonna  so this means so you start with something z not it sits inside some larger complex manifold  z and you have to choose so a compact we have the diffeomorphism Phi choose compact set  k 1 and what we achieved was approximating this Phi on this particular compact case but  this Phi is going to do something arbitrary outside here right so this means that let's  see so now it had maybe a psi 1 or something so this is not really good you have to correct  it so you have to do something like considering okay now you rather want psi 1 inverse composed  with Phi because if you compose this with this you're back in business so now you try  to approximate this and then correct your original approximation and then you try to  set up inductive scheme to exhaust all of the not but of course you have to take care  that your sequence of maps also converges to a diffeomorphism of C to N and that makes  it a little bit tricky but that's basically the idea and I think actually that I am done  so thank you.
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