This article collects problems and solutions from the course: Modern Geometry, Spring 2025 by Siye Wu of National Tsing Hua University. The topic of the course is on group actions on manifolds and equivariant cohomology.

Problem X.

The unit -sphere has a symplectic form which is the standard area form, normalised so that its total area is . The circle group acts on by standard rotations around the axis through the north and south poles. Now let be equipped with a symplectic form from the factors in an obvious way. The diagonal action of on is Hamiltonian and the additive constant in the moment map is fixed by the requirement that the average of on is . Find the push-forward to of the Liouville measure on by the moment map using two methods: first directly and then by the Duistermaat-Heckman formula. [Hint: The Duistermaat-Heckman formula expresses the Fourier transform of the push-forward measure in terms of the fixed-point data. The inverse Fourier transform of the contribution from each fixed point is ill-defined, but can be changed to either or .]

Solution.

Let with each equipped with the standard area form of total area . Denote by the symplectic form on the -th copy of , normalized so that We consider the diagonal -action on given by simultaneous rotation about the -axis on each sphere. On each factor , with coordinates , this circle action has moment map Hence the diagonal moment map on is We endow with the product symplectic form so that the Liouville volume form is We wish to compute the pushforward measure on . Equivalently, we seek a density on supported in so that We start from computing the pushforward measure.

1. Pushforward on a single sphere: On one sphere with coordinates , the moment map , , has level sets which are circles of latitude at height . In standard spherical coordinates with , the area form is Writing , we have . Hence . Consequently, for any test function , However, this expression already integrates out the angle but has not accounted for the Jacobian of the change of variable. A more direct check is: Equivalently, the pushforward measure on has constant density . In particular, the total mass is , as desired.

2. Product manifold and diagonal moment map: On we write coordinates on the first sphere and on the second, so and The diagonal moment map is . We view as a function , and integrate out the angle variables .

   Fix a value . The fiber is On that fiber, one has . The volume element can be written as But along , . Thus A more direct way is to compute: First integrate over (they range over independently), giving a factor . Then one integrates over . Equivalently, perform the -integral via the Dirac delta measure: That indicator is nonzero if and only if Hence As a function of , one checks Therefore, putting everything together, Consequently, One easily checks the total mass is , which equals .

Now we derive the result again via Duistermaat-Heckman and inverse Fourier transform:

1. Since is the product measure on , and , we have By the localization formula we have Therefore

2. We now recover the density by It is a well-known standard Fourier-transform: Hence We conclude again:

Problem XI.

Let be a connected Lie group with Lie algebra . Consider a coadjoint orbit . Each determines a function by Also, each defines a vector field on , induced by the infinitesimal coadjoint action of on .

1. Show that and verify directly that

   Now define the Kirillov–Kostant 2-forms by

2. Show that are well-defined, non-degenerate, and closed. Thus are both symplectic forms on .

3. Show that the -action on is Hamiltonian with respect to both , and that the moment maps are given by Hint: Show that .

4. Verify directly that where denote the Poisson brackets associated to the symplectic forms , respectively.

Solution.

1. Recall that the vector field on is for all . Then for any and , we have

   Note that we identify since is linear. Then we have, also, ’s are linear in ’s by Jacobi identity.¹

2. For well-definedness and non-degeneracy, since , it suffice to check the first component.

   • (well-defined) For any and , we have, note that ’s are linear in ’s,

   • (non-degenerate) If for all , then we have for all and . This means for all , thus and hence .

   To show is closed, recall that for any -form , we have Thus we compute

   The last line vanishes by Jacobi identity. Thus are symplectic forms on .

3. To show the action is Hamiltonian, it suffices to check and, for the corresponding moment map , for all .

   • We claim that . Indeed, for all , we have Then for Lie derivative we have Thus consider for all we obtain the moment map of .

   • The Lie derivative of the moment map reads Meanwhile, for all . We conclude .

   Now, for each , we have We conclude for all .

4. Recall the Poisson bracket where , etc. Note that , hence Then the Poisson bracket of the moment map is,

Problem XII.

1. Consider the defining representation of on . For any integer , let be the th symmetric power of , which is also a representation of . A maximal torus of is the subgroup consisting of the diagonal matrices in . Find the weight space decomposition of according to representations of .

2. It is known that the complexification of is . Let be the subgroup of consisting of lower-triangular matrices. Then acts on holomorphically by right multiplication, and is a complex space. Show that is holomorphically diffeomorphic to , and that there is a holomorphic left -action on .

3. For , let be the 1-dimensional representation of on given by Consider the holomorphic line bundle over . Show that the action of on the space of holomorphic sections of defines a representation of that is equivalent to its representation on .

   Hint: Holomorphic sections of are identified with holomorphic functions such that

Solution.

We note that the symmetric tensor product has the isomorphism , the space of homogeneous polynomials of degree , induced by and with the standard basis of . Then the representations are defined by this isomorphism.

1. The maximal torus of reads hence its Lie algebra reads The representation of on is explicitly, for ,

   This means admits the weight space decomposition with weights .

2. Let be a homogeneous coordinate on . We define the map

   where the action of on is given by the defining representation of , where Note that this is clearly a holomorphic action by quotient topology.

   We show the map is a biholomorphism: It is a holomorphic map since it descends from the holomorphic action , .

   • Well-defined: For all , since where we have for all .

   • 1-1 and onto: For all with , let , we have . Meanwhile, let , we have . Thus is onto.

     On the other hand, if , say for some nonzero . Then we have where is the solution of , which has unique solution since . Then we conclude , hence the map is 1-1.

   • Finally, has a holomorphic inverse given by where . This is well-defined on because for any solutions of , say and , we have which has a unique solution for any fixed .

3. We first inspect the structure of the line bundle

   Define the character Then form the associated line bundle where for , .

   Local trivializations over the standard charts.

   Cover by the two affine charts: On , write . Choose a holomorphic lift which satisfies . Define a local frame Any element of the fiber of over can be written uniquely as for some . Thus trivializes on : a general section restricts to

   On , write . Choose a holomorphic lift which satisfies . Define Then any element of the fiber over can be written as for a unique .

   Transition function on .

   On the overlap , both coordinates and make sense. We need to compare the two frames and . By construction, Hence there exists a unique such that We claim . Indeed, while To match these, multiply on the right by . One finds Since , we conclude that in the identification of frames, Indeed, , so . Thus the transition function of from to is .

   Global holomorphic sections of .

   A global section is given by a pair of holomorphic functions on and satisfying the compatibility Thus must be a polynomial of degree at most . Writing one has which is holomorphic at . Hence there is a one‐to‐one correspondence between sections of and complex polynomials in of degree . Equivalently, in homogeneous coordinates, a homogeneous polynomial of total degree . Therefore so .

   -equivariance.

   We are left to show that the representations are equivalent.

   • The action on in terms of the homogeneous polynomials is explicitly as follows: Recall is spanned by the monomials and for the standard (defining) action on is Hence on homogeneous degree‐ polynomials one sets In particular, each basis monomial transforms by

   • The action of on reads as follows: Over the standard affine chart with , every section can be written as where and is a polynomial of degree . Now let We wish to compute explicitly in the chart . Recall the action of on , it follows that on the affine coordinate , the point is sent (by ) to Denote

     Recall that Under , this frame transforms by for some . Equivalently, on one finds a unique lower‐triangular matrix such that A direct computation shows On the other hand, Hence We must have i.e.  Equating entries yields From and , we get One checks consistency with and . Thus In particular, Since , it follows that where Hence

     paragraph4. Action on a general section . Given the ‐action is Substitute the formula for : Thus But is merely the local frame at the new point . Hence the new section in the ‐trivialization is In other words, if corresponds (in ) to the polynomial , then Equivalently, in homogeneous coordinate form, on is These two descriptions agree under the inhomogeneous identification , since

   We conclude they are indeed isomorphic representations.