On the Steinberg map and Steinberg cross-section for a symmetrizable indefinite Kac-Moody group (Q2715667)

Let \(G\) be a symmetrizable indefinite Kac-Moody group over \({\mathbf C}\) and let \(Tr_{\Lambda_1}, \ldots, Tr_{\Lambda_{2n-l}}\) be the characters of the fundamental irreducible representation of \(G\), defined as convergent series on a certain part \(G^{{\text{tr-alg}}} \subseteq G\). Following Steinberg in the classical case and Brüchert in the affine case, the author defines the Steinberg map \(\chi := (Tr_{\Lambda_1}, \ldots, Tr_{\Lambda_{2n-l}})\) as well as the Steinberg cross section \(C\), together with a natural parametrization \(\omega: {\mathbf C}^n{\times}({\mathbf C}^{\times})^{n-l} \to {\mathbf C}\). The author investigates the local behaviour of \(\chi\) on \(C\) near \(\omega((0,\ldots,0){\times}(1,\ldots,1))\), and he shows that there exists a neighborhood of \((0,\ldots,0){\times}(1,\ldots,1)\), on which \(\chi{\circ}\omega\) is a regular analytical map, satisfying a certain functional identity. This identity has its origin in an action of the center of \(G\) on \(C\).