In the literature, they often extend the Standard Model by adding a so-called vector-like fermion which is a multiplet invariant under $SU(2)_L\times U(1)$. The left- and right-handed components of vector-like multiplet are supposed to transform the same way, under the symmetry group. As a result, their mass terms, are not forbidden by the symmetry. See for an example see these slides.
I have problem understanding the gauge transformation. If the multiplet $\Psi$ is invariant under $ SU(2)_L$, only the left-handed particles $\Psi_L =P_L \Psi$ should be charged under the gauge symmetry. The right-handed component $\Psi_R$ should not be charged under $ SU(2)_L$.
So my understanding is that vector-like multiplet should be in a representation of $SU(2)$ that is different from the usual chiral representation of the SM fermion doublets.
If this is correct, then how is that vector-like representation of fermions related to the SM chiral representation? Is there a linear transformation?
Any helps or comments would be appreciated.