Vol. 2, No. 2, 2008

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A topological property of quasireductive group schemes

Najmuddin Fakhruddin and Vasudevan Srinivas

Vol. 2 (2008), No. 2, 121–134

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 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.

group scheme, quasireductive, nearby cycle
Mathematical Subject Classification 2000
Primary: 14L15
Secondary: 20G35
Received: 12 March 2007
Revised: 25 July 2007
Accepted: 25 August 2007
Published: 15 March 2008
Najmuddin Fakhruddin
School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba
Vasudevan Srinivas
School of Mathematics
Tata Institute of Fundamental Research
Homi Bhabha Road, Colaba