Generalized Flag Varieties of Kac-Moody Groups

Abstract

Let $\mathcal{G}$ be a Kac—Moody group and ${\mathcal{P}_Y} \subset \mathcal{G}$ any parabolic subgroup. The aim of Section 7.1 is to realize the homogeneous space ${\mathcal{X}^Y}{\text{: = }}\mathcal{G}{\text{/}}{\mathcal{P}_Y}$ as a projective ind-variety so that the Schubert subvarieties $\mathcal{X}_Y^w \subset \mathcal{G}{\text{/}}{\mathcal{P}_Y}$ are indeed closed finite-dimensional (projective) irreducible subvarieties. Fix a (dominant integral) weight $\lambda \in {D_\mathbb{Z}}$ such that, for $1 \leqslant i \leqslant \ell ,\lambda (\alpha _i^ \vee ) = 0$ iff i∈Y. Such a λ is called Y-regular. Let V (λ) be an integrable highest weight g-module with highest weight λ. From the last chapter, V (λ) acquires a $\mathcal{G}$-module structure.