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.

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 |