Index reduction formulas for twisted flag varieties. II
type: Item
instance of: scholarly article
title: Index reduction formulas for twisted flag varieties. II
zbMATH Open document ID: 0918.16013
DOI: 10.1023/A:1007793218556
author: Ivan Panin
author: A. R. Wadsworth
author: Alexander Merkurjev
published in: \(K\)-Theory
publication date: 1998-10-18T00:00:00Z
review text: The present paper is the sequel to Part I [K-Theory 10, No. 6, 517-596 (1996; Zbl 0874.16012)] by the same authors, referred to as [I]. The twisted flag varieties considered in these papers are projective varieties over an arbitrary field \(F\) on which an adjoint semisimple algebraic group \(G\) acts, the action being transitive after scalar extension to a separable closure \(F_s\) of \(F\). In [I], a general formula for the change in Schur index of an arbitrary central simple \(F\)-algebra when scalars are extended to the function field of a twisted flag variety \(X\) was given. It involves various central simple algebras \(A_G(\psi)\) attached to the elements \(\psi\) in the character group of the center of the simply connected cover of \(G_s=G\times F_s\) as well as integers \(n_{\psi,P,E}\) depending on \(\psi\), on the parabolic subgroup \(P\) of \(G_s\) such that \(X\times F_s\simeq G_s/P\) and on certain finite extensions \(E\) of \(F\). In [I], this formula was made explicit for \(G\) simple, classical and of inner type, by a case-by-case determination of the algebras \(A_G(\psi)\) and the integers \(n_{\psi,P,E}\). The purpose of the present paper is to give similar explicit formulas in the cases not covered by [I]. The authors discuss the cases where \(G\) is a classical simple group of outer type or an exceptional simple group. They also show how to derive index reduction formulas for the flag varieties associated to a product \(G_1\times G_2\) from those for \(G_1\) and \(G_2\), and for the flag varieties of transfers (Weil restriction of scalars). Thus, index reduction formulas for all flag varieties for all adjoint semisimple groups can be obtained by combining the results in [I] with those of the present paper. Since this paper was written, alternative proofs of some of its results, avoiding a case-by-case analysis, have been found by \textit{A. S. Merkurjev} [Doc. Math. 1, 229-243 (1996; Zbl 0854.20060), J. Ramanujan Math. Soc. 12, No. 1, 49-95 (199…
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 19E08
zbMATH DE Number: 1197102
zbMATH Keywords: central simple algebras
zbMATH Keywords: Brauer groups
zbMATH Keywords: semisimple linear algebraic groups
zbMATH Keywords: Borel varieties
zbMATH Keywords: twisted flag varieties
zbMATH Keywords: projective varieties
zbMATH Keywords: change of Schur index
zbMATH Keywords: function fields
zbMATH Keywords: groups of inner type
zbMATH Keywords: parabolic subgroups
zbMATH Keywords: index reduction formula
zbMATH Keywords: Brauer-Severi varieties
reviewed by: Jean-Pierre Tignol
MaRDI profile type: Publication
full work available at URL: a:1007793218556
OpenAlex ID: W2094042668
description: scientific article; zbMATH DE number 1197102