Affine Demazure weight polytopes and twisted Bruhat orders

Marc Besson, Sam Jeralds and Joshua Kiers

Abstract

For an untwisted affine Kac–Moody Lie algebra \(\mathfrak{g}\) with Cartan and Borel subalgebras \(\mathfrak{h} \subset \mathfrak{b} \subset \mathfrak{g}\), affine Demazure modules are certain \(U(\mathfrak{b})\)-submodules of the irreducible highest-weight representations of \(\mathfrak{g}\). We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the \(\mathfrak{h}\)-weights of such a module. Using methods of geometric invariant theory, we determine inequalities which define these polytopes; these inequalities come in three distinct flavors, specified by the standard, opposite, or semi-infinite Bruhat orders. We also give a combinatorial characterization of the vertices of these polytopes lying on an arbitrary face, utilizing the more general class of twisted Bruhat orders.

Keywords: Affine Demazure module, twisted Bruhat order, geometric invariant theory, affine Lie algebras.

AMS Subject Classification: Primary 14M15; secondary 22E66, 17B67, 05E10, 52A40.