Index reduction formulas for twisted flag varieties. I
The varieties considered in this important paper 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 \(F\). The authors give a precise formula for the change in Schur index of arbitrary central simple \(F\)-algebras, when scalars are extended from \(F\) to the function field of such a variety. Only the case where \(G\) is of inner type is considered in detail in this paper, but the same methods also apply to groups of outer type, which the authors intend to treat in the second part. If \(X\) is a twisted flag variety as above, then there is a parabolic subgroup \(P\) of \(G_s=G\times F_s\), uniquely determined by \(X\) up to conjugacy, such that \(X\) becomes isomorphic to \(G_s/P\) over \(F_s\). If \(G\) is of inner type, the general index reduction formula for an arbitrary central simple \(F\)-algebra \(D\) then takes the form \(\text{ind}(D\otimes F(X))=\gcd(n_\psi\text{ind}(D\otimes A(\psi)))\), where \(\psi\) runs over the characters of the center of the simply connected cover of \(G_s\), and the integers \(n_\psi\) and the central simple \(F\)-algebras \(A(\psi)\) depend only on \(\psi\). In the final sections of the paper, this formula is made explicit by a thorough description of the integers \(n_\psi\) and the algebras \(A(\psi)\) for the various types of groups \(G\) and parabolic subgroups \(P\). The authors thus recover various special cases previously obtained. The first of these special cases is due to \textit{A. Schofield} and \textit{M. Van den Bergh} [Trans. Am. Math. Soc. 333, No. 2, 729-739 (1992; Zbl 0778.12004)], who used Quillen's computation of the group \(K_0\) of Brauer-Severi varieties to obtain an index reduction formula for scalar extension to the function field of a Brauer-Severi variety. The present paper uses the same technique, relying on Panin's computation of the \(K\)-theory of twisted flag varieties. (Also submitted to MR).
adjoint semisimple linear algebraic groups
scientific article; zbMATH DE number 947549
scientific article