The paper under review is concerned with the interesting problem of finding generators for rational loop groups, which arises from dressing actions and various geometric applications. The idea to study this problem, introduced by Terng and Uhlenbeck, is to find the so-called simple elements, i.e. generators with smallest number of poles. In previous work, Donaldson, Fox and the author found generators for rational loop groups of all classical groups and \(G_2\) with the reality condition given by a compact real form.
In this paper, the author considers the above problem from another direction, namely, searching for generators of rational loop groups with the reality condition given by a noncompact real form. The author successfully solves the case \(\mathrm{GL}(n,\mathbb{C})\) with a noncompact real form \(\mathrm{GL}(n,\mathbb{R})\). As a starting point, in the absence of the eality condition, the problem is easier and the author shows that \(\mathrm{GL}(n,\mathbb{C})\)-loops can be generated by two types of elements \[ p_{\alpha,\beta,V,W}(\lambda)=\frac{\lambda-\alpha}{\lambda-\beta}\pi_V +\pi_W, \] where \(\mathbb{C}^n=V\oplus W\), and \[ m_{\alpha, k, N}=Id + \left(\frac{1}{\lambda-\alpha}\right)^k N, \] where \(k\) is positive and \(N\) is two-step nilpotent. It turns out that the result after imposing the reality condition is still quite elegant: \(\mathrm{GL}(n,\mathbb{C})\)-loops with the \(\mathrm{GL}(n,\mathbb{R})\)-reality condition can be generated by the above simple loops satisfying the reality condition, together with products of two simple loops of the form \(p_{\alpha,\beta, V, W}\) which do not satisfy the reality condition.
Apart from this key result, the author also considers nilpotent dressing with simple poles as well as higher order poles, and makes applications to ZS-AKNS flows. Another application is towards the \(n\)-dimensional systems associated to \(\mathrm{GL}(n,\mathbb{R})/O(n)\).