# Finding symmetry of a part of an equation, given the group transformation property of another part

I am reading this paper on Dyons and Duality in $\mathcal{N}=4$ super-symmetric gauge theory. The author finds the zero modes or a dirac equation obtained by considering first order perturbations to the Bogolomony equation for bps monopoles. He finds out that when the simple root used for embedding the SU(2) monopole is simple then the bosonic zero modes which I believe is the solution for the higgs field transform as $1 \oplus 0$, and when it is not a simple root it transforms as two doublets. Hope this is not required for my question.

He founds it in the following way. The fermionic zero modes are the solution to $\psi$ given by $$\psi = \chi \otimes (1\,\,\, 0) - i\sigma_2 \xi^{\star}\otimes(0\,\,\,1)$$ The zero modes are eigenvectors of $J=j+s$ the sum of the spatial rotation, and global gauge transformation. $\psi$ transforms as a $J \otimes \frac {1}{2}$ representation.

My question is if the zero modes of $\psi$ is a triplet of SU(2) how, do I conclude that zero modes of $\chi$ transforms as $1 \oplus 0$ wrt to $J$, as stated in the paper? Also, if $\psi$ is a doublet, how do the zero modes of $\chi$ transform,

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