Demazure character formula for semi-infinite flag varieties

TL;DR

This paper proves projective normality of Schubert varieties in semi-infinite flag varieties and connects Demazure modules, Weyl modules, and Macdonald polynomials through geometric realizations.

Contribution

It establishes the projective normality of semi-infinite Schubert varieties and links algebraic modules to geometric spaces, providing new interpretations.

Findings

Schubert varieties are projectively normal

Demazure modules correspond to global sections of semi-infinite Schubert varieties

Geometric realizations of Weyl modules and Macdonald polynomials

Abstract

We provide a proof that every Schubert variety of a semi-infinite flag variety is projectively normal. This gives us an interpretation of a Demazure module of a global Weyl module of a current Lie algebra as the (dual) space of the space of global sections of a semi-infinite Schubert variety. Moreover, we give geometric realizations of Feigin-Makedonskyi's generalized Weyl modules, and the $t = \infty$ specialization of non-symmetric Macdonald polynomials.