On unitary groups modulo infinitesimals (Q5917357)

Let \(D\) be a skew field of characteristic not two with an involution \(*\) and a valuation \(\omega\) such that \(\omega (x^*) = \omega(x)\). Denote by \(R\) the ring of this valuation and by \(J\) its maximal ideal. Consider a finite-dimensional vector space \(V\) over \(D\) with a non-degenerate Hermitian form \(x.y\) satisfying the Cauchy-Schwarz inequality in the form \(2\omega (u.v) \geq \omega(u.u) + \omega(u.v)\) for \(u,v \in V\) [cf. \textit{M. Chacron}, Pac. J. Math. 119, 1-87 (1985; Zbl 0575.20046)]. A linear transformation \(\varphi\) of \(V\) is said to be unitary \(\text{mod }1 + J\) if it preserves lengths modulo the group \(1 + J\) of 1-units. These transformations form a group \(C\) which the author studies here. For each value \(g \geq 0\) define a subgroup \(C_g\) of \(C\) as the set of all \(1 + \varphi\) such that \(2\omega(u.v \varphi) \geq \omega(u.u) + \omega(v.v) + 2g\) for \(u\), \(v\) in \(V\) and not both zero. These \(C_g\) are shown to form a chain of \(*\)-closed normal subgroups of \(C\) intersecting in \(\{1\}\) and satisfying \([C_g, C_h] \subseteq C_{g+h}\). Moreover \(C_g \neq \{1\}\) provided that \(V\) is not 1-dimensional. If one gives up the Cauchy-Schwarz inequality and assumes that \(V\) is even- dimensional, an orthogonal sum of hyperbolic planes, the author describes a condition for a matrix to lie in \(C\) which he uses to obtain a generating set for \(C\) in this case.