REDUCTIVE SUBGROUP SCHEMES OF A PARAHORIC GROUP SCHEME

Abstract

Let K be the field of fractions of a complete discrete valuation ring $\mathcal{A}$ with residue field k, and let G be a connected reductive algebraic group over K. Suppose $\mathcal{P}$ is a parahoric group scheme attached to G. In particular, $\mathcal{P}$ is a smooth affine $\mathcal{A}$-group scheme having generic fiber $\mathcal{P}$_K = G; the group scheme $\mathcal{P}$ is in general not reductive over $\mathcal{A}$.

If G splits over an unramified extension of K, we find in this paper a closed and reductive $\mathcal{A}$-subgroup scheme $\mathcal{M}\subset \mathcal{P}$ for which the special fiber $\mathcal{M}$_k is a Levi factor of $\mathcal{P}$_k. Moreover, we show that the generic fiber $M={\mathcal{M}}_{\mathrm{K}}$ is a subgroup of G which is geometrically of type C(μ) – i.e., after a separable field extension, M is the identity component $M={C}_G^o\left(\phi \right)$ of the centralizer of the image of a homomorphism ϕ: μ_n → H, where μ_n is the group scheme of n-th roots of unity for some n ≥ 2. For a connected and split reductive group H over any field $\mathcal{F}$, the paper describes those subgroups of H which are of type C(μ).