Newton polyhedra and the converse of the Borel-Weyl theorem (Q789500)

Let \({\mathfrak g}\) be a complex semisimple Lie algebra, \({\mathfrak b}\) its Borel subalgebra, \({\mathfrak h}\) a Cartan subalgebra of \({\mathfrak g}\), \({\mathfrak n}=[{\mathfrak b},{\mathfrak b}]\), W the Weyl group of the pair (\({\mathfrak g},{\mathfrak h})\). A \({\mathfrak b}\)-module V \((\dim_{{\mathbb{C}}}V<\infty)\) is said to be semisimple if it is a semisimple \({\mathfrak h}\)-module. Denote by \({\mathbb{C}}_{\lambda}\), \(\lambda\in {\mathfrak h}^*\!_{{\mathbb{C}}}\) the 1- dimensional \({\mathfrak b}\)-module such that \({\mathfrak n}{\mathbb{C}}_{\lambda}=0\), \(hv=\lambda(h)v\) for any \(v\in {\mathbb{C}}_{\lambda}\), \(h\in {\mathfrak h}\). The author proves the following theorem. Let V be a semisimple \({\mathfrak b}\)-module, then \((a)\quad \dim H^*({\mathfrak n},V)\geq | W|. (b)\quad \dim H^*(n,V)=| W| \quad \Leftrightarrow \quad V\simeq {\mathbb{C}}_{\lambda}\otimes U\) for some \(\lambda\in {\mathfrak h}^*\!_{{\mathbb{C}}}\), where the \({\mathfrak b}\)- module U has the structure of an irreducible \({\mathfrak g}\)-module.