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We prove the so-called Unitary Isotropy Theorem, a result on isotropy of a unitary involution. The analogous previously known results on isotropy of orthogonal and symplectic involutions as well as on hyperbolicity of orthogonal, symplectic, and unitary involutions are formal consequences of this theorem. A component of the proof is a detailed study of the quasi-split unitary grassmannians.

Abstract. In the present article we investigate properties of the category of the integral Grothendieck-Chow motives over a field.We discuss the Krull-Schmidt principle for integral motives, provide a complete list of the generalized Severi-Brauer varieties with indecomposable integral motive, and exploit a relation between the category of motives of twisted flag varieties and integral p-adic representations of finite groups.

Motivic equivalence for algebraic groups was recently introduced by De Clercq [Compos. Math. 153 (2017), pp. 2195–2213], where a characterization of motivic equivalent groups in terms of higher Tits indexes is given. As a consequence, if the quadrics associated to two quadratic forms have the same Chow motives with coefficients in F 2 \mathbb {F}_2 , this remains true for any two projective homogeneous varieties of the same type under the orthogonal groups of those two quadratic forms. Our main result extends this to all groups of classical type, and to some exceptional groups, introducing a notion of critical variety. On the way, we prove that motivic equivalence of the automorphism groups of two involutions can be checked after extending scalars to some index reduction field, which depends on the type of the involutions. In addition, we describe conditions on the base field which guarantee that motivic equivalent involutions actually are isomorphic, extending a result of Hoffmann on quadratic forms.

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