I have only recently started exploring this region of homogeneous spaces and its geometry and the question is born from that and given the beginner state of my exploration the questions might sound ill-framed. 


All this is motivated by trying to understand how the spectrum of the Dirac Operator is obtained on homogeneous spaces which in physics is a very natural situation. 


1. Can anyone explain to me how connections on SU(2) can be labeled by the irreducible representations of SU(2) (what in physics is called "spin") ?
2. Accordingly how are laplacians on SU(2) labelled by irreducible representations of SU(2)?
3. Literature seem to state that symmetric, traceless and divergence less tensors on $S^3$ are also labeled by irreducible representations of SU(2). How is that?
4. What is the precise definition of a "tensor harmonic" on a homogeneous space? 
And how do "tensor harmonics" on SU(2) give a basis for expanding sections of the spin-bundle on SU(2)?
5. SU(2) can be thought of as the homogeneous space $SU(2)\times SU(2)/SU(2)$ under the diagonal action and then how does sections of this principle bundle give a basis for tensor valued functions on SU(2)?
6. I am adding another point here. In the above bundle with the diagonal action the projection map to the base space is $(g\_1,g\_2) \mapsto g\_1g\_2 ^{-1}$ which will have as inverse images orbits of the above diagonal action. And hence one sees the fibers of this principle bundle structure of this homogeneous space. 


Now some calculations in the literature seem to tell me that choosing a section in this bundle with respect to the above projection map somehow canonically defines me vielbiens in the base $SU(2)$. 


The point being that given the usual diagonal metric on $S^3$ one could have guessed the vielbiens upto a sign. But somehow it seems that the choice of a section fixes the signs. 


Can anyone kindly explain how the choice of a section over $S^3$ (the fields obviously lying in the bundle-space $SU(2)\times SU(2)$) gives a "natural" vielbien on the base $S^3$? 


One of the popular sections here are called "thermal sections" in physics.



Representations of SU(2) and tensors on SU(2) 
You may wish to take a look at this [survey article](http://geometrie.math.uni-potsdam.de/documents/baer/dirac_spectrum_survey.pdf) by Christian Bär about the spectrum of the Dirac operator. It contains a section (1.1.2.2) on homogeneous spaces. For the particular case of three dimensions, there is also his [earlier paper](https://doi.org/10.1007/BF01199016).


