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A topological property of quasireductive group schemes. (English) Zbl 1156.14325

Summary: In a recent paper, G. Prasad and J. Yu [J. Algebr. Geom. 15, No. 3, 507–549 (2006; Zbl 1112.14053)] introduced the notion of a quasireductive group scheme \(G\) over a discrete valuation ring \(R\), in the context of Langlands duality. They showed that such a group scheme \(G\) is necessarily of finite type over \(R\), with geometrically connected fibres, and its geometric generic fibre is a reductive algebraic group; however, they found examples where the special fibre is nonreduced, and the corresponding reduced subscheme is a reductive group of a different type. In this paper, the formalism of vanishing cycles in étale cohomology is used to show that the generic fibre of a quasireductive group scheme cannot be a restriction of scalars of a group scheme in a nontrivial way; this answers a question of Prasad, and implies that nonreductive quasireductive group schemes are essentially those found by Prasad and Yu.

MSC:

14L15 Group schemes
20G35 Linear algebraic groups over adèles and other rings and schemes

Citations:

Zbl 1112.14053
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