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.