I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as
$$ \mathcal L=-\frac {1}{4} \text{tr} (F_{\mu \nu} F^{\mu \nu}) + \overline \psi (i \tilde D-m)\psi $$
with
$$ \tilde D=\gamma^\mu D_\mu \qquad D_\mu=\partial_\mu-igA_\mu $$
As far as i understood, in this equation $\psi$ is an element of a representation $V$ of $\mathfrak g$ and $A_\mu \in \text{End}(V)$, right? What's puzzling me now, is that this equation has to be Lorentz-invariant as well, so the $\psi$'s are still Dirac spinors, that is they are elements of the fundamental representation $\Delta$ of the Clifford Algebra $\text{Cliff}(1,3)$. How should i interpret this ambiguity and in particular how should i calculate $\gamma^\mu A_\mu \psi$ in local coordinates?
The field $\psi$ is not one Dirac spinor but a collection of those: it belongs to tensor product representation $V\otimes\Delta$, and accordingly has two indices $\alpha$, $a$. In components (I guess this is what you mean by local coordinates) one has $$(\gamma^{\mu}A_{\mu}\psi)_a^{\alpha}=\gamma_{ab}^{\mu}A_{\mu}^{\alpha\beta}\psi^b_{\beta},$$ where $a,b$ are spinor labels and $\alpha,\beta$ are gauge group representation labels.
For example, when $V$ is the fundamental representation of $SU(3)$, the multicomponent field $\psi$ contains three Dirac spinors corresponding to different quark colors.