User david petrecca - MathOverflow

Answer by David Petrecca for Finding spherical representations of $GL(n, \mathbb{C})$.

2013-05-14T14:10:13Z

There is a paper by Kramer about pairs $(G, K)$ with $G$ connected Lie group and $K$ spherical in $G$ that is for all irreducible representations of $G$, the space of vectors fixed by $K$ is at most 1-dimensional.

If I recall correctly Kramer gives some propreties of spherical pairs and provides the full classifications of spherical pairs $(G, K)$ with $G$ compact and simple.

Here: link text

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Normalized Hamiltonian holomorphic vector fields on Sasakian manifolds

2013-01-22T18:26:40Z

Hello, I am reading the paper Futaki; Ono; Wang Transverse Kähler geometry of Sasaki manifolds and toric Sasaki-Einstein manifolds. J. Differential Geom. 83 (2009), no. 3, 585–635.

For your convenience I repeat the setting. Let $(M^{2n+1}, \eta, \xi, g)$ be a compact Sasakian manifold and $\pi_\alpha: U_\alpha \rightarrow C^n$ the submersions defining the characteristic foliation on $M$.

Hamiltonian holomorphic vector fields are introduced to be complex vector fields $X$ on $M$ such that the field $d\pi_\alpha X$ is holomorphic and the function $u_X := i \eta(X)$ is such that $\iota_X \frac 1 2 d\eta = i \bar \partial_B u_X$.

Let $h$ be a real basic function such that $\rho^T - (2n+2)\omega^T = i \partial_B \bar \partial_B h$ (transverse Ricci form and transverse Kaehler form).

They consider normalized vector fields. They are the ones such that $\int_M u_X e^h (1/2 d\eta)^n \wedge \eta = 0$.

It can be checked that the set of Hamiltonian holomorphic vector fields is closed under the Lie bracket.

Is it true also for normalized ones?

After using Theorem 5.1 of the paper which states that the space of normalized ham holo flds is in correpspondance with the $2n+1$ eigenspace of the operator on complex valued basic functions $$ \Delta^h u = \Delta u - \nabla^i u \nabla_i h $$ where $\Delta$ is the basic laplacian and $\nabla$ the transverse LC connection.

The correspondence is given by $u \mapsto u \xi + \nabla^i u (\partial_{z_i} - \eta(\partial_{z_i})\xi)$. Btw this is where I have another question. Shouldn't it be $u \mapsto -iu \xi + ...$ in order to have a hamiltonian holomorphic vector field in the above sense?

Anyway I computed the bracket of two fields $X, Y$ image of basic functions $u,v$ and found out that the Hamiltonian function of $[X,Y]$ is $Xv - Yu = \nabla^i u \nabla_i v - \nabla^i v \nabla_i u$. Its itegral with respect to the weighted measure is then $$ \int (\Delta^h u \cdot v - u \Delta^h v)e^h (1/2 d\eta)^n \wedge \eta $$ which is not the self-adjoint relation for $\Delta^h$, which would contain some conjugates, being $\Delta^h$ the $\bar \partial$-Laplacian wrt the weighted measure. This is where I am stuck.

thank you

David

Answer by David Petrecca for Group or manifold ?

2013-02-21T11:38:18Z

I'm not sure what you mean by the star product but yes, you see a quotient G/H as the set of cosets. There are topological requirements on H though, for instance being closed etc... at least for G/H to be a differentiable manifold, I'm not sure about topological manifold.

Finally, the idea to prove the sphere statement is to notice that U(n) acts transitively on a sphere of appropriate dimension, choose a point and prove that its stabilizer is U(n-1) somehow seen inside U(n) (e.g. lower block of a matrix in U(n))

You can find more details on both my answers in the Lie Groups chapter in Warner's book (Foundations of diff geom I think)

David

Write homogeneous spherical space forms as coset spaces

2011-11-26T19:42:22Z

Hi,

Let $S=G/K$ be a sphere written as coset space. I know there are just few possibilities for $G$, and $K$ due to the classification of compact connected groups that can be transitive on a sphere.

If $F \subset G$ is a finite fixed point free subgroup, can we write $S/F = G / (F \times K)$??

This should be true for real projective spaces, but for example how about lens spaces $S^{2n-1}/\mathbb Z_q$?

thanks

David

Comment by David Petrecca

2013-05-14T19:07:22Z

I thought it could be related to the question in the other discussion (which apparently was merged with this one). It wasn't even clear to me you meant the same thing for spherical representations :)

Comment by David Petrecca

2013-04-04T07:58:58Z

Thank you, Craig! When you talk about the splitting of the Lie algebra you mean a sasakian analog of the decomposition of the Lie algebra of holomorphic fields that holds in the Kahler case? In this case for the algebra of normalized ones ofc. Anyway without using it is a subalgebra I already managed to write a decomposition by using that it is a set of representatives for the quotient {Ham. holo flds}/ $\xi$, in analogy with other splittings holding in the Sasaki-extremal case and more in general for transvesely Kahler foliations (Tondeur-Nishikawa)

Comment by David Petrecca

2011-11-28T21:37:19Z

Sorry for adding a comment as answer. Anyway it is enough to have $F$ (finite) normal (hence cenral) in $G$, or just central to well-define a $G$-action on $M/F$, but what about other sufficient cases? For instance, if we see $S^{2n-1} = U(n)/U(n-1)$ and $F$ is the cyclic group acting by multiplying by $e^{2 \pi i/q}$ on each complex coordinates, of course $F$ is central in $G$ and we can say that $S^{2n-1}/F$ is homogeneous under $U(n)$. How about a $SO(2n)$-action?

Comment by David Petrecca

2011-11-26T23:23:59Z

And in the general case what if $F$ and $K$ commute? For example for finite subgroups of $SU(2)$