If we do the Legendre transform to the Lagrangian , we will swithc from to where is the generalized momentum and the importance about this change of variables is that sits in a more symmetric position to . Indeed, the Lagrange’s equation will be equivalent to Hamilton’s equation (The Legendre transform is involutive, so we can recover Lagrangian from Hamiltonian; though see here for the discussion that hamiltonian may not be convex).
How to start from symplectic geometry and produce Hamilton’s equations: The idea is Darboux’s theorem, which states in a small neighbourhood around any point on there exist suitable local coordinates s.t. the symplectic form becomes . The form induces a natural isomorphism of the tangent space with the cotangent space (since it is closed 2-form). If (think of at each time the function gives the energy at each point of ), then it gives rise to a vector field defined by . To see that this corresponds to Hamilton’s equation, note that and , see Hamiltonian vector field for reference.
We list two properties which is immediate:
A Hamiltonian is constant along the integral curve of . More generally, if have zero Poisson brackets, then is constant along the integral curve of and vice versa (the abstract form of Noether’s theorem).
The Hamiltonian flow preserves the symplectic form (Liouville’s theorem).
Next we want to motivate moment map. We start with a Lie group acting by symplectomorphism on . Intuitively is the phase space and elements of are symmetries. The idea of a moment map is that it gives an embedding of the Lie algebra of infinitesimal symmetries into the Poisson algebra . See this answer for more details. See this for how it corresponds to the usual momentum.
The moment map is also connected to Noether’s theorem, in the sense that if the action preserves the Hamiltonian , then for every infinitesimal the image under (the dual) of the moment map is a constant of motion. See here for more details.
Motivation for spherical varieties: They are essentially the simplest kind of -varieties, in the sense that there are only finitely many -orbits for all its -equivariant birational models. This is equivalent to having a dense open -orbit for a Borel , and also equivalent to some multiplicity-free condtions if it is quasi-projective, see here (where the motivation is to find subgroup that acts locally transitively on the flag variety ) and here.
Chevalley gives us an isomorphism for connected reductive. Let denote the composite . For generic , we can look at the irreducible components of . More precisely set . This space is no longer irreducible, but acts transitively on the components. Let be one of them. Then let , i.e. is the Galois group of . The principal result is that is always a crystallographic reflection group.
Now we insert some motivations behind flag varieties. The idea is that Borel-Weil-Bott theorem essentially says that every finite-dimensional irreducible representation arises from parabolic induction. Quoted from this mathoverflow answer,
Taken by itself, the Borel-Weil theorem provides a somewhat concrete geometric model using line bundles on the flag variety for all finite dimensional irreducible representations of a (complex or compact) semisimple Lie group. Up to isomorphism these representations are parametrized by “dominant” characters of a maximal torus. The existence was originally an indirect consequence of work by E. Cartan and then Weyl, but the actual representations are not easy to write down. Instead, some indirect information about characters (or weight space multiplicities) was cleverly developed.
The Beilinson–Bernstein localization is a power generalization of this idea. It says we have an equivalene of categories between -modules and . The LHS can be thought of as deformation of finite-dimensional -representation with a fixed central character corresponding to the weight .
horospherical type: first invariant is
If , then the cotangent bundle of has a right action of and the irreducibel compnent of , but the irreducible compnents are just , so in this case it is the trivial
Horospherical subgroup is horospherical if contains a maximal unipotent subgroup of , if we take the normalizer of then it is parabolic subgroup, and also contains . Then is a torus. Proof: Let be a rep of stablizer of a line is , decompose it into dominant characters, write where . The fact that means that are higheset weight vectors and note that the intersection of stablizer of lines generated by are parabolic. We note that is the set of that actually stablize for all .
Consider , can view as living over but also acts on on the right in adiition to . If we consider the action of . There is a map , it is called -mondromic differential operators. Take , then we get a map to . By Harish chandra isomorphism we can identify
There is such that if we invert in then is surjective. The reduction is that the rational is surjective (because is a finite -module where .
Let be the sheaf of differential operators on , the pushforward has an action by . Define to be the -monodromic operators. The global section of is .
If is a homomorphism (the base field), we can form where is the base field regarded as a module over the universal envelopping algebra induced by . Since still has an action of , if we take global section then there is a map . The theorem is if is dominant, then this map is surjective.
in the case of Borel, this is the Harish-Chandra character. The argument is similar to Bernstein-Saito polynomial by passing to the generic fiber and use some finite genenration result.
Brion-Vust (local structure theorem): There is a dense open and a parabolic (containing a given Borel) and a levi such that is -equivariantly isomorphic to for some -variety , where where and acts trivially on .
Corollary: On the open , the subgroups are conjugate in . If is spherical, this is automatic. Pick one . Then there exists unique homospherical such that (we have ).
Example: Let the variety of nondegenerate symplectic forms. This is already spherical
Pick a generic point which is the standard basis. Let be the stablizer of in , it looks like two blocks of (which is 6 dimensional, the dimension of ). The is . The should be contained in and satisfies , so just take . The associated is .
For , we should take to be the stabliezer of the flag , theb is the diagonal torus with first and last entry the same and second and third the same.
Describe where the decomposition of .
We get a projection and dualize we get injection . The kernel of the projection is so the outcome is . If we fix a Cartan involution then we can produce a maximal commutative subalgebra of . Let be the moment map image closure. In general doesn’t have irreducible generic fiber. Let be the normalization of in . The map is finite and is integral (i.e. has irreducible generic fiber). Quotient by we get . The goal is we want to produce a finite group such that . Knopp prove the map is canonical. Let ( should be thought of as quotient of ). By the universal property of integral closure factorize through . Main result: There is a unique subgroup such that . Often true that
Example: If , we can take where so that . The Weyl group is that of .
https://physics.stackexchange.com/questions/351903/simple-explanation-of-why-momentum-is-a-covector