Abstract

AbstractWe compute the spaces of sections of powers of the determinant line bundle on the spherical Schubert subvarieties of the Beilinson–Drinfeld affine Grassmannians. The answer is given in terms of global Demazure modules over the current Lie algebra.

Highlights

  • Let be a simple complex Lie algebra

  • In this paper we are interested in the representation theory of two natural infinite-dimensional analogues of the Lie algebra – the current algebra [ ] = ⊗ C[ ] and the affine Kac–Moody Lie algebra with the natural embedding [ ] ⊂

  • ◦ identify the dual sections of the determinant line bundle on symmetrised BD spherical Schubert varieties with global Demazure modules

Read more

Summary

Introduction

Let be a simple complex Lie algebra. To simplify the notation, in the introduction we assume that is laced. L, ) – corresponding to a collection of dominant integral nonzero weights ∈ + and an integer l > 0; in particular, if all are fundamental and l = 1, one gets back the global Weyl module (this is no longer true in the case that is not laced). ◦ identify the dual sections of the determinant line bundle on symmetrised BD spherical Schubert varieties with global Demazure modules. The reason is that the central role in our paper is played by the spherical Schubert varieties in the affine (Beilinson–Drinfeld) Grassmannians These varieties are naturally labelled by coweights (rather than weights), which explains our choice of notation. Generalities We start by describing the notation for the key objects of the paper

Classical objects
Current algebra modules
Affine Lie algebras and Demazure modules
Affine Grassmannians
Beilinson–Drinfeld Schubert varieties
The highest-weight algebras
BD Grassmannians and spherical Schubert varieties
Sections of the determinant line bundle
Embeddings of the BD Schubert varieties
Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call