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 $$ \dim(Z(N)/Z(N)\cap G(\mathcal{O}))=\dim(M/M\cap g(\mathcal{O}))=\frac{1}{2}(\nu_F(\det(k|M\cap g(\mathcal{O}))-\nu_F(\det(k|M))) $$ Here $k$ is the killing form. Why are those two equalities true?
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1$\begingroup$ It may be simplest here to ask Roman B. himself (?); he can usually be reached by email even if he's not at MIT. By the way, that paper is here: intlpress.com/site/pub/pages/journals/items/mrl/content/vols/… $\endgroup$– Jim HumphreysCommented Aug 10, 2017 at 0:34
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