On quasi-reductive group schemes

By Gopal Prasad, J. Yu. 2004

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This paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpre- tation of the dual group (as a Chevalley group over Z) of a reductive group. We define a quasi-reductive group over a discrete valuation ring R to be an affine flat group scheme over R such that (i) the fibers are of finite type and of the same dimension; (ii) the generic fiber is smooth and connected, and (iii) the identity component of the reduced special fiber is a reductive group. We show that such a group scheme is of finite type over R, the generic fiber is a reductive group, the special fiber is connected, and the group scheme is smooth over R in most cases, for example when the residue characteristic is not 2, or when the generic fiber and reduced special fiber are of the same type as reductive groups. We also obtain results about group schemes over a Dedekind scheme or a Noetherian scheme. We show that in residue characteristic 2 there are non-smooth quasi-reductive group schemes with generic fiber SO2n+1 and they can be classified when R is strictly Henselian.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/66a41dcc2ed8b361df5affaaa38a0037cd6e2c03

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Geometric Langlands duality and representations of algebraic groups over commutative rings (2004, methodology): “By theorem 1.2 of [ PY ], the group e GZp is reductive if the following conditions hold:”