For an untwisted affine Kac-Moody Lie algebra g with Cartan and Borel subalgebras h Ă b Ă g, affine Demazure modules are certain U pbq-submodules of the irreducible highest-weight representations of g. We introduce here the associated affine Demazure weight polytopes, given by the convex hull of the 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.
For G a reductive group and T Ă B a maximal torus and Borel subgroup, Demazure modules are certain B-submodules, indexed by elements of the Weyl group, of the finite irreducible representations of G. In order to describe the T -weight spaces that appear in a Demazure module, we study the convex hull of these weights -the Demazure polytope. We characterize these polytopes both by vertices and by inequalities, and we use these results to prove that Demazure characters are saturated, in the case that G is simple of classical Lie type. Specializing to G " GL n , we recover results of Fink, Mészáros, and St. Dizier, and separately Fan and Guo, on key polynomials, originally conjectured by Monical, Tokcan, and Yong.
In this note we explicitly construct top-dimensional components of the cyclic convolution varieties. These components correspond (via the geometric Satake equivalence) to irreducible summands, where N ≥ 1 and β is a positive root. Furthermore, we deduce from these constructions a nontrivial lower bound on the multiplicity of these subrepresentations when β is not a simple root. In an appendix, we contrast this approach with a combinatorial proof of the same results using Littelmann paths.
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