Codimensions of Root Valuation Strata

The topic of this paper arises naturally in the context of affine Springer fibers, which we now take a moment to discuss. Let G be a semisimple complex algebraic group, and let g denote its Lie algebra. We then have the affine Grassmannian X = G(F )/G(O), where O is the ring C[[ǫ]] of formal power series, and F is its fraction field C((ǫ)). For any u ∈ g(F ) = g ⊗C F the closed subset X = {g ∈ G(F )/G(O) : Ad(g)u ∈ g(O) = g ⊗C O}