REDUCTIVE SUBGROUP SCHEMES OF A PARAHORIC GROUP SCHEME
| Autor: | George J. McNinch |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Algebra and Number Theory
Image (category theory) 010102 general mathematics Field of fractions Reductive group 01 natural sciences Separable space Combinatorics Group scheme Field extension Algebraic group 0103 physical sciences 010307 mathematical physics Geometry and Topology Identity component 0101 mathematics Mathematics |
| Zdroj: | Transformation Groups. 25:217-249 |
| ISSN: | 1531-586X 1083-4362 |
| DOI: | 10.1007/s00031-018-9508-3 |
| Popis: | 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). |
| Databáze: | OpenAIRE |
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