Let $G$ a semisimple simply connected group over $\mathbb{C}$. Let $\gamma\in G(\mathbb{C}[[t]])$ such that $\gamma$ is regular semisimple on $G(\mathbb{C}((t)))$.
We consider $I_{\gamma}$ the group scheme over $\mathbb{C}[[t]]$ which is the centralizer of $\gamma$. In particular, on the generic fiber, we know that it's a torus, but has no reason to be smooth or flat on $\mathbb{C}[[t]]$.
We have a canonical section of this group scheme given by the unity $e$.
My question is what is the valuation of $e^{*}\Omega^{1}_{I_{\gamma}/\mathbb{C}[[t]]}$ also known as the Neron defect of smoothness?
More precisely can we rely this valuation to the discriminant of $\gamma$?
