REDUCTIVE SUBGROUP SCHEMES OF A PARAHORIC GROUP SCHEME

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: μn → H, where μn 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).