Fakhruddin, Najmuddin; Srinivas, Vasudevan A topological property of quasireductive group schemes. (English) Zbl 1156.14325 Algebra Number Theory 2, No. 2, 121-134 (2008). 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 Keywords:group scheme; quasireductive; nearby cycle Citations:Zbl 1112.14053 PDFBibTeX XMLCite \textit{N. Fakhruddin} and \textit{V. Srinivas}, Algebra Number Theory 2, No. 2, 121--134 (2008; Zbl 1156.14325) Full Text: DOI