This is an expansion which frequently occurs in the papers of Camporesi and Higuchi but I couldn't find a derivation of it in either their review papers or in standard books on homogenous vector bundles.

Consider the lie group $G$ and its subgroup $H$ and a homogeneous vector bundle be defined on $G/H$ corresponding to a representation $\tau$ of $H$ and $\sigma$ be a section of the principle bundle formed by $G$ over $G/H$. Let $\psi$ be a section of this homogeneous vector bundle and $\psi ^a $ be the component of the section along some basis labeled by $a$. So $a$ runs through all natural numbers till the dimension of $\tau$. This basis is defined at every fiber of the bundle by pushing forward by $\sigma$ a chosen basis in the fiber at identity of $G/H$.

Then $\psi ^a$ has the following expansion known as the "Harmonic Expansion",

$\psi ^{a}(x) = \sum _{\lambda} \sum _I \sum _{\xi} U^{\lambda}((\sigma (x))^{-1})^{a\xi}_{I} \psi ^{I}_{\lambda \xi}$

where $\lambda$ runs over representatives from all equivalence classes of representations of $G$ which when restricted to the subgroup $H$ of $G$ contain $\tau$ in them. $I$ runs over the dimension of $\lambda$ and $\xi$ runs over natural numbers till the multiplicity of $\tau$ in the above restriction. $U^{\lambda}(g)^{a\xi}_{I}$ is the matrix entry of the $\lambda$ representation of $g$ between the basis field $I$ (of the vector space on which $\lambda$ is) and $a\xi$ (basis vector labeled by $a$ in the copy $\xi$ of the representation $\tau$ )

I would like to know the derivation of the above expansion, especially how apparently the Frobenius Reciprocity theorem plays role in it?

And in this expansion which is the "harmonic"?

By "harmonic" we should have a quantity here which is an eigen-function of some laplacian. Which is it here and of which laplacian?

*Like to put in one more aspect here,*

In a homogeneous vector bundle on $G/H$, $G$ naturally has a representation on the space of sections of the homogeneous bundle. It seems that sections in the irreducible representations of this are also called "Harmonics".

What is the rationale for this terminology?