A TOPOLOGICAL PROPERTY OF QUASI-REDUCTIVE GROUP SCHEMES

Abstract. In a recent paper, Gopal Prasad and Jiu-Kang Yu introduced the notion of a quasi-reductive 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 non-reduced, 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 quasi-reductive group scheme cannot be a restriction of scalars of a group scheme in a non-trivial way; this answers a question of Prasad, and implies that non-reductive quasi-reductive group schemes are essentially those found by Prasad and Yu. In a recent paper [9], Gopal Prasad and Jiu-Kang Yu introduced the notion of a quasi-reductive group scheme over a discrete valuation ring R: this is an affine, flat group scheme pi: G → SpecR, such tha