Index reduction formulas for twisted flag varieties. II
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 (1997; Zbl 0906.16006)]. Even if the approach in these papers is more direct, the thorough discussion of various special cases presented here is undoubtedly useful. Particularly noteworthy are the careful summary exposition of transfers given in section 7 and the description of Tits algebras for groups of type \(E_6\) and \(E_7\) in section 8. (Also submitted to MR).
scientific article; zbMATH DE number 1197102