Modules with Demazure Flags and Character Formulae

Abstract

In this paper we study a family of finite-dimensional graded representations of the current algebra of sl₂ which are indexed by partitions. We show that these representations admit a flag where the successive quotients are Demazure modules which occur in a level ℓ-integrable module for A₁¹ as long as ℓ is large. We associate to each partition and to each ℓ an edge-labeled directed graph which allows us to describe in a combinatorial way the graded multiplicity of a given level ℓ-Demazure module in the filtration. In the special case of the partition 1s and ℓ=2, we give a closed formula for the graded multiplicity of level two Demazure modules in a level one Demazure module. As an application, we use our result along with the results of Naoi and Lenart et al., to give the character of a g-stable level one Demazure module associated to B¹n as an explicit combination of suitably specialized Macdonald polynomials. In the case of sl₂, we also study the filtration of the level two Demazure module by level three Demazure modules and compute the numerical filtration multiplicities and show that the graded multiplicites are related to (variants of) partial theta series.