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Working paper

Reduced arc schemes for Veronese embeddings and global Demazure modules

arxiv.org. math. Cornell University, 2019. No. 1912.07988.
Dumanski, I., Feigin E.
We consider the projective arc schemes of the Veronese embeddings of the flag varieties for simple Lie groups of type ADE. The arc schemes 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 of 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 localization 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 $A_1$ we give the precise list of generators of the reduced arc scheme structure of the Veronese curves. In general type we introduce the notion of the global higher level Demazure modules and identify the graded components of the homogeneous coordinate rings with these modules.