A topological property of quasireductive group schemes

Abstract

In a recent paper, Gopal Prasad and Jiu-Kang Yu 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 etale 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.