Summary: [For the entire collection see Zbl 0741.00064.]
Let \(A=\bigl({2\atop -2}{-2\atop 2}\bigr)\), \(\mathfrak g\) the associated Kac- Moody Lie algebra, and \(G\) the Kac-Moody group. Let \(P\) be a maximal parabolic subgroup in \(G\), with associated fundamental weight \(\omega\). Let \(W\) (resp. \(W_ P\)) be the Weyl group of \(G\) (resp. \(P\)). For \(\tau\in W/W_ P\), let \(X(\tau)\) be the associated Schubert variety in \(G/P\). Let \(V_ \omega\) be the integrable, highest weight module (over \(\mathbb{C}\)) with highest weight \(\omega\). Fix a highest weight vector \(e\) in \(V_ \omega\). Let \(e_ \tau=\tau e\), \(V_ \tau=U^ +_ \mathbb{Z} e_ \tau\otimes K\), where \(K\) is the base field and \(U^ +_ \mathbb{Z}\) is the \(\mathbb{Z}\)-subalgebra of the universal enveloping algebra of \(\mathfrak g\), generated by \(\{X^ n_ \alpha/n!,\alpha\) a simple root}. Let \(L\) be the tautological line bondle on \(\mathbb{P}(V_ \tau)\). For the canonical embedding \(X(\tau)\hookrightarrow \mathbb{P}(V_ \tau)\), we denote the restriction of \(L\) to \(X\) by just \(L\).
In this paper, we construct explicit bases for \(H^ 0(X(\tau),L^ m)\), \(m\in Z^ +\) and obtain as consequences the normality of \(X(\tau)\), a character formula for \(H^ 0(X(\tau),L^ m)\) etc. We also obtain similar results for Schubert varieties in \(G/B\). The results have been announced in C. R. Acad. Sci., Paris, Sér. A I 305, 183-185 (1987; Zbl 0632.22012).