Simplified presentations and embeddings of Demazure modules

TL;DR

The paper addresses embedding higher-level Demazure modules for untwisted affine Lie algebras into tensor products of lower-level Demazure modules, connecting to classical finite-dimensional decompositions in the anti-dominant limit. It introduces three auxiliary modules with simplified presentations and proves a general embedding theorem for $r$-admissible weight splittings $\boldsymbol{\mu}\in \mathcal{P}(\mu,k)$, yielding $\mathbf{D}_{\mu}^{rk} \hookrightarrow \mathbf{D}_{\mu_1}^{r} \otimes \cdots \otimes \mathbf{D}_{\mu_k}^{r}$. It extends the CV13 simplifications beyond the $\mathfrak{g}$-stable case and links these algebraic embeddings to crystal-theoretic constructions for Demazure crystals and classical decompositions with respect to maximal semisimple subalgebras. The results provide a framework for constructing explicit Demazure-crystal models via connected components of tensor products of lower-level crystals and have implications for graded limits in quantum affine algebras.

Abstract

For an untwisted affine Lie algebra we prove an embedding of any higher level Demazure module into a tensor product of lower level Demazure modules (e.g. level one in type A) which becomes in the limit (for anti-dominant weights) the well-known embedding of finite-dimensional irreducible modules of the underlying simple Lie algebra into the tensor product of fundamental modules. To achieve this goal, we first simplify the presentation of these modules extending the results of \cite{CV13} in the $\mathfrak{g}$-stable case. As an application, we propose a crystal theoretic way to find classical decompositions with respect to a maximal semi-simple Lie subalgebra by identifying the Demazure crystal as a connected component in the corresponding tensor product of crystals.

Theorems & Definitions (28)

• Remark
• Theorem 1
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