Algebraic loop groups and moduli spaces of bundles.

*(English)*Zbl 1020.14002The concept of an algebraic loop group emerged in the context of conformal quantum field theory and its algebro-geometric framework, especially in connection with moduli spaces of bundles over curves, the theory of infinite Grassmannians, Krichever theory and infinite-dimensional Lie algebras. Whilst many basic properties of algebraic loop groups are well-explored in characteristic zero, comparatively little has been known in positive characteristic, at least so until recently.

The paper under review changes this situation radically, in that it starts an attempt to check the known facts on algebraic loop groups (in characteristic zero) in full generality. In the seven sections of this article, the author develops a general theory of algebraic loop groups, affine Grassmannians, generalized Schubert varieties, and moduli stacks of \(G\)-bundles over projective curves in characteristic \(p> 0\). The main results include a proof of the normality and Cohen-Macaulay property of the generalized Schubert varieties introduced here, the construction of line bundles on the affine Grassmannian, and a proof that the latter ones induce line bundles on the moduli stack of \(G\)-bundles. Throughout this important, generalizing and systematizing work, the author expertly points out the relations of his constructions to the various other ones (in characteristic zero) as well as to some linked recent results in positive characteristic.

The paper under review changes this situation radically, in that it starts an attempt to check the known facts on algebraic loop groups (in characteristic zero) in full generality. In the seven sections of this article, the author develops a general theory of algebraic loop groups, affine Grassmannians, generalized Schubert varieties, and moduli stacks of \(G\)-bundles over projective curves in characteristic \(p> 0\). The main results include a proof of the normality and Cohen-Macaulay property of the generalized Schubert varieties introduced here, the construction of line bundles on the affine Grassmannian, and a proof that the latter ones induce line bundles on the moduli stack of \(G\)-bundles. Throughout this important, generalizing and systematizing work, the author expertly points out the relations of his constructions to the various other ones (in characteristic zero) as well as to some linked recent results in positive characteristic.

Reviewer: Werner Kleinert (Berlin)

##### MSC:

14D20 | Algebraic moduli problems, moduli of vector bundles |

14L40 | Other algebraic groups (geometric aspects) |

14M15 | Grassmannians, Schubert varieties, flag manifolds |