Degree 3 relative invariant for unitary involutions

Abstract

Using the Rost invariant for non split simply connected groups, we define a relative degree $3$ cohomological invariant for pairs of orthogonal or unitary involutions having isomorphic Clifford or discriminant algebras. The main purpose of this paper is to study general properties of this invariant in the unitary case, that is for torsors under groups of outer type ${\sf A}$. If the underlying algebra is split, it can be reinterpreted in terms of the Arason invariant of quadratic forms, using the trace form of a hermitian form. When the algebra with unitary involution has a symplectic or orthogonal descent, or a symplectic or orthogonal quadratic extension, we provide comparison theorems between the corresponding invariants of unitary and orthogonal or symplectic types. We also prove the relative invariant is classifying in degree $4$, at least up to conjugation by the non-trivial automorphism of the underlying quadratic extension. In general, choosing a particular base point, the relative invariant also produces absolute Arason invariants, under some additional condition on the underlying algebra. Notably, if the algebra has even co-index, so that it admits a hyperbolic involution, which is unique up to isomorphism, we get a so-called {\em hyperbolic} Arason invariant. Assuming in addition the algebra has degree $8$, we may also define a {\em decomposable} Arason invariant. It generally does not coincide with the hyperbolic Arason invariant, as the hyperbolic involution need not be totally decomposable.