Generic norms. II (Q1901782)

[For part I cf. Contemp. Math. 131, Pt. 2, 587-603 (1992; Zbl 0776.17023).] Let \(A\) be a central simple algebra (always finite-dimensional over an infinite field \(F\)). An involution of the first kind of \(A\) (i.e. fixing \(F\)) is one of two types: orthogonal or symplectic, depending on whether over a splitting field for \(A\), the involution reduces to transposition or to a symplectic involution. With such an involution \(A\) necessarily has even degree \(2m\) say. For an involution \(J\) of symplectic type on \(A\) the author defined a generic norm \(n_H\) on the special Jordan algebra \(H (J,A)=(1+J) A\). If \(K\) is an involution of orthogonal type on \(A\), then \(S(A,K)=(1-K) A\) is a Lie algebra; one can always write \(K=i_u J\) where \(J\) is an involution of symplectic type and \(i_u\) is the inner automorphism induced by a skew symmetric element \(u\). The linear maps \(a\to ua\) and \(a\to au^{-1}\) are bijections between \(H(A, J)\) and \(S(A,K)\). The author proves that for a central simple algebra \(A\) of degree \(2m\) with involution \(K\) of orthogonal type there exists a polynomial function \(n_k\) of degree \(m\) on \(S(A,K)=(1- K)A\) and a polynomial map \(p_K\) of degree \(m-1\) of \(S(A,K)\) into itself such that \(bp_K (b)=p_K (b)b=n_K(b)1\). These functions \(p_K\) and \(n_K\) which are homogeneous and unique up to a scalar factor are called generic adjoint and generic norm respectively. They satisfy the identity \[ p_K (p_K (b))=(-1)^m n_K (b)^{m-2} n_A (u^{-1}) b, \tag{1} \] where \(u\) is an invertible \(K\)-skew element and \(n_A\) the generic norm in \(A\). An algebra with involution \((A,J)\) is said to split if it can be decomposed into a tensor product of factors stable under \(J\) and \(\neq F\). When \(A\) is central simple of degree 4 and \(J\) is symplectic, it is shown that \(A\) always splits. For an involution \(K\) of orthogonal type the author shows that \((A,K)\) is trivial. Here \(\delta (K)\) is defined as the coset \((\text{mod } F^{\times2})\) of \(n_A (u)^{-1}\), where \(u\) is the \(K\)-skew element occurring in (1). This provides a short proof of results of \textit{M. A. Knus}, \textit{R. Parimala} and \textit{R. Sridharan} [J. Indian Math. Soc., New Ser. (to appear)].