Invariant fields of symplectic and orthogonal groups

The projective orthogonal and symplectic groups POn(F) and PSpn(F) have a natural action on the F vector space V′=Mn(F) Mn(F). Here we assume F is an infinite field of characteristic not 2. If we assume there is more than one summand in V′, then the invariant fields F(V′)POn and F(V′)PSpn are natural objects. They are, for example, the centers of generic algebras with the appropriate kind of involution. This paper considers the rationality properties of these fields, in the case 1, 2, or 4 are the highest powers of 2 that divide n. We derive rationality when n is odd, or when 2 is the highest power, and stable rationality when 4 is the highest power. In a companion paper joint with Tignol, we prove retract rationality when 8 is the highest power of 2 dividing n. Back in this paper, along the way, we consider two generic ways of forcing a Brauer class to be in the image of restriction.