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One is familiar from Quantum Theory that each of the angular momentum generators $L_{x,y,z}$ are Killing Fields for the standard metric on $S^2$ and the sum of the squares of these generators gives the Laplacian on R^3.

It seems from some literature that this idea in some sense generalizes.

Vaguely what it seems to me is that for a homogeneous spaces $G/H$ if $K_i$ are the killing fields (on $G/H$ ?) then $\sum_i K_i K_i$ is the Laplacian on $G/H$.

It would be helpful if one can tell me what is the precise statement that contains the above idea and also what are the caveats and the proof of why it should be so.

In this context people also talk of the "Casimir Laplacian". What precisely is that?

Casimir Laplacian comes about in this way,

If $T_a$ happen to be the Killing Fields on $G$ and $X_b$ be the Killing fields on $H$ then in some cases (when the algebra is reductive?) a relation of this kind holds,

$$K_i K_i = T_aT_a - X_b X_b$$

(sum over repeated indices implied)

Here too I don't know the precise statement or the proof, but just am seeing allusions to it in the papers.

There is also this issue of 3 `different' ways of defining the laplacian, either as the ordinary one $\nabla ^{\mu}\nabla _{\mu}$

$$or$$

as $\sum_i \nabla_{X_i^*}\nabla _ {X_i ^*}$ where $\nabla _ {X_i ^*}$ is defined as the so called $H$-connection on $G/H$."

This is how apparently the $H$ connection's evaluation on a section $\psi: G/H \rightarrow G$ along the vector filed $X$ (section of a homogeneous vector bundle over $G/H$) is defined,

$$\nabla_{X^{*}} \psi (x_{0}) = lim_{t\rightarrow 0} \frac{exp(-tX)\psi(\gamma_X(t))-\psi(x_0)}{t}$$

where $\gamma_X(t) = exp(tX)x_o$ is the integral curve of $X$ through the origin of $G/H$ i.e $x_0$

$$or$$

as through the Lie derivative as $\sum_i L_{K_i} L_{K_i}$

How to understand the difference and the relations between these notions of Laplacians?


As an example of the kind of relationships I am trying to understand let me quote $3$ of such equations,

  • $$\nabla_{\beta} \nabla ^{\beta} V^{\alpha} = \left ( \sum _i L_{K_i}L_{K_i} V \right )^{\alpha} + R^{\alpha} _{\beta} V^{\beta} + f^{\beta}_{\gamma}{^{\alpha}} \nabla _{\beta} V^{\gamma}$$

where the structure constant $f$ and the Ricci Tensor are of the $G/H$ and $V$ is a vector field on $G/H$ (written here in the vielbein basis) and the connection is on $G/H$ but the Killing fields are of $G$. The Ricci Tensor can for such spaces be written in terms of the structure constants or the Casimir operator of the representation of $H$ which defines the vector bundle in concern.

  • $$\sum _i L_{K_i}L_{K_i} = -\sum _{\lambda} C_2(\lambda)$$ where the right hand side is a sum over Casimirs of all irreducible representations of $G$ and the Lie derivatives on $G$ acting in a `natural' way on the fields of $G/H$

  • For the `H-connection" the first equation reduces to,

    $$\nabla_{\beta} \nabla ^{\beta} V^{\alpha} = \left ( \sum _i L_{K_i}L_{K_i} V \right )^{\alpha} - (f_p f^{p})^{\alpha}_{\gamma} V^{\gamma} $$

    where the $f$ are the generators of the representation of $H$. Basically the new terms is the components of the Casimir of that representation of $H$ along the $G/H$ components.

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2 Answers

This is standard, at least for homogeneous manifolds $G/H$ where $G$ is compact and semisimple. See for instance the paper The laplacian on homogeneous spaces.

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Thanks for the paper reference. Basically I am trying to understand the equation which this referred paper of yours states in equation $(5)$. But he has a reference for where that can be found. I have added some examples of the relationships that I had in mind when I first put up the questions. These are what I am after. Any reference along that direction will be a great help. –  Anirbit Mar 2 '10 at 8:45
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This answer is mainly based on appendix B4 of the review article: Harmonic analysis and propagators on homogeneous spaces by R. Camporesi treating Laplacians on homogeneous spaces. I think that the exposition in this article is quite transparent and systematic. Here, only the semisimple case is considered.

The definition of a Casimir Laplacian acting on sections of a vector bundle is given by: $\sum_{ij} C^{ij} L_{K_i} L_{K_j}$, where $C$ is the Cartan-Killing metric and $L_{K_j}$ is the Lie derivative on the sections. The sum is over all the generators of $G$. This definition is quite general, because the Lie derivative can be defined on tensors (sections of tensor products of the tangent bundle), forms (sections of the antisymmetric tensor products of the cotangent bundle), sectios of line bundles, half forms, spinors etc.

Considering first the case of vector fields (sections of the tangent bundle). In Camporesi's article, the relation of the Casimir Laplacian to some other Laplacians on $G/H$ is explained for the case of vector fields:

Two connections on the tangent bundle of $G/H$ are explicitely constructed: The Levi-Civita connection and the H-connection (given explicitely in appendix A) which is not torsion-free. The connection Laplacian of the Levi-Civita connections is the usual Laplace-Beltrami Laplacian. The Laplace-Beltrami Laplacian is not in general diagonal on spaces of sections belonging to an irreducible representation of $G$, while the connection Laplacian based on the H-connection is diagonal which makes the H-connection based Laplacian more appropriate in harmonic analysis.

This fact can be seen from their explicit expression given in the article, while the Laplace-Beltrami Laplacian differs from the Casimir Laplacian by a variable term, the H-connection based Laplacian differs by a constant term.

In fact on spaces of sections belonging to an irreducible representation of highest weight $\lambda$ it is equal to: $C_2(\lambda)-C_2(\tau)$. Where $\tau$ is the isotropy representation (the generally reducible representation corresponding to the action of $H$ on the tangent space).

This property of the H-connection is general for homogeneous vector bundles over $G/H$. In this case, $\tau$ is the inducing representation of $H$ defining the vector bundle and the H-connection is the natural connection on this bundle.

An important special case is the Laplacian of line bundles on flag manifolds. In this case, the line bundle is characterized by a weight $\lambda$ of $G$ and the eigenvalue of the H-connection Laplacian on a space of section carrying the representation of a highest weight $\Lambda$ of $G$ is $C_2(\Lambda)-C_2(\lambda)$. When, $\lambda$ is positive and $\Lambda = \lambda$ the eigenvalue becomes zero, and this happens exactly on the space of holomorphic sections which is a manifestation of the Borel-Weil theorem.

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Can you kindly edit the LaTeX errors which are not rendering properly? It is in the very first definition and hence making your argument hard to follow. And I stumbled upon this whole idea from this very paper that you have referred to and what I was reading. :) –  Anirbit Mar 15 '10 at 10:08
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