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The flag manifold of Kac-Moody Lie algebra. (English) Zbl 0764.17019
Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 161-190 (1989).
[For the entire collection see Zbl 0747.00038.]
The author constructs the flag variety of a Kac-Moody Lie algebra as an infinite-dimensional scheme. There are several constructions by Kac- Peterson, Kazhdan-Lusztig, S. Kumar, O. Mathieu, P. Slodowy and J. Tits, but there the flag variety is understood as a union of finite-dimensional varieties.
Here two methods of construction of the flag variety are given. For a Kac-Moody Lie algebra $${\mathfrak g}$$, let $$\hat{\mathfrak g}$$ be the completion of $${\mathfrak g}$$. The first construction is to realize the flag variety as a subscheme of Grass $$(\hat{\mathfrak g})$$, the Grassmann variety of $$\hat{\mathfrak g}$$. More precisely, taking the Borel subalgebra $${\mathfrak b}_ -\subset\hat{\mathfrak g}$$ and regarding this as a point of Grass$$(\hat{\mathfrak g})$$, we define the flag variety as its orbit by the infinitesimal action of $$\hat{\mathfrak g}$$ in Grass$$(\hat{\mathfrak g})$$.
The other construction is to realize the flag variety as $$G/B_ -$$. Of course, in the Kac-Moody Lie algebra case, we cannot expect that there is a group scheme whose Lie algebra is $${\mathfrak g}$$. But we can construct a scheme $$G$$ on which $${\mathfrak g}$$ acts infinitesimally from the left and the right. Then we define the flag variety $$G/B_ -$$, where $$B_ -$$ is the Borel subgroup. More precisely, we consider the ring of regular functions as in V. G. Kac and D. H. Petersen [Prog. Math. 36, 141-166 (1983; Zbl 0578.17014)]. Then its spectrum admits an infinitesimal action of $${\mathfrak g}$$. But its action is not locally free. Roughly speaking, $$G$$ is the open subscheme where $${\mathfrak g}$$ acts locally freely (Proposition 6.3.1). The flag variety of a Kac-Moody algebra shares the similar properties to the finite-dimensional ones, such as Bruhat decompositions.

##### MSC:
 17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 14M15 Grassmannians, Schubert varieties, flag manifolds 14L15 Group schemes