Reduced arc schemes for Veronese embeddings and global Demazure modules

TL;DR

This paper studies the structure of arc spaces associated with Veronese embeddings of flag varieties, revealing their connection to Demazure modules and introducing global higher level Demazure modules for types beyond A1.

Contribution

It demonstrates that graded components of coordinate rings are cocyclic modules over current algebras and introduces global higher level Demazure modules as a generalization.

Findings

Each graded component is a cocyclic module over the current algebra.

The fiber at the special point is isomorphic to an affine Demazure module.

In type A1, explicit generators of the reduced arc space are provided.

Abstract

We consider arc spaces for the compositions of Pluecker and Veronese embeddings of the flag varieties for simple Lie groups of types ADE. The arc spaces are not reduced and we consider the homogeneous coordinate rings of the corresponding reduced schemes. We show that each graded component of a homogeneous coordinate ring is a cocyclic module over the current algebra and is acted upon by the algebra of symmetric polynomials. We show that the action of the polynomial algebra is free and that the fiber at the special point of a graded component is isomorphic to an affine Demazure module whose level is the degree of the Veronese embedding. In type A$_1$ (which corresponds to the Veronese curve) we give the precise list of generators of the reduced arc space. In general type, we introduce the notion of global higher level Demazure modules, which generalizes the standard notion of the global Weyl modules, and identify the graded components of the homogeneous coordinate rings with these modules.