Maximal indexes of Tits algebras

Abstract

Let $G$ be a split simply connected semisimple algebraic group over a field $F$ and let $C$ be the center of $G$. It is proved that the maximal index of the Tits algebras of all inner forms of $G_L$ over all field extensions $L/F$ corresponding to a given character $\chi$ of $C$ equals the greatest common divisor of the dimensions of all representations of $G$ which are given by the multiplication by $\chi$ being restricted to $C$. An application to the discriminant algebra of an algebra with an involution of the second kind is given.