I have a few questions with respect to Bezrukavnikov's proof of the dimension formula for affine springer fibers in Fixed point set on affine flag manifolds. The setting is as follows: Let G be a reductive group over $\mathbb{C}$. Let $F=\mathbb{C}((t))$ with the valuation that satisfies $\nu_F(t)=1$, and $\mathcal{O}=\mathbb{C}[[t]]$. Let $N\in g(\mathcal{O})$ be a regular semisimple nil-element. Let $Z(N)$(resp $z(N))$ be the centralizer of $N$ in $G(F)$(resp $g(F)$). Let $M$ be the lattice of integral elements in $z(N)$, here $x\in g(F)$ is integral means that for any $Q$ an ad-invarant polynomial of $g$, $Q(x)\in \mathcal{O}$. Then $$dim(Z(N)/Z(N)\cap G(\mathcal{O}))=dim(M/M\cap g(\mathcal{O}))=1/2(\nu_F(det(k|M\cap g(\mathcal{O}))-\nu_F(det(k|M)))$$ Here $k$ is the killing form. Why are those two equalities true?