Level Zero Fundamental Representations over Quantized Affne Algebras and Demazure Modules

Abstract

Let $W({\varpi }_k)$ be the finite-dimensional irreducible module over a quantized affne algebra $U_q^{\prime }(\mathfrak{g})$ with the fundamental weight ${\varpi }_k$ as an extremal weight. We show that its crystal $B(W({\varpi }_k))$ is isomorphic to the Demazure crystal $B^{-}(-{\Lambda }_0+{\varpi }_k)$. This is derived from the following general result: for a dominant integral weight $\lambda$ and an integral weight $\mu$, there exists a unique homomorphism $U^{-}q(\mathfrak{g})(u_{\lambda }\otimes u_{\mu })\to V(\lambda +\mu )$ that sends $u_{\lambda }\otimes u_{\mu }$ to $u_{\lambda +\mu }$. Here $V(\lambda )$ is the extremal weight module with $\lambda$ as an extremal weight, and $u_{\lambda }\in V(\lambda )$ is the extremal weight vector of weight $\lambda$.