Level Zero Fundamental Representations over Quantized Affine Algebras and Demazure Modules

Let W (̟k) be the finite-dimensional irreducible module over a quantized affine algebra U ′ q(g) with the fundamental weight ̟k as an extremal weight. We show that its crystal B(W (̟k)) is isomorphic to the Demazure crystal B −(−Λ0 +̟k). This is derived from the following general result: for a dominant integral weight λ and an integral weight μ, there exists a unique homomorphism Uq(g)(uλ⊗ uμ) → V (λ+μ) that sends uλ ⊗ uμ to uλ+μ. Here V (λ) is the extremal weight module with λ as an extremal weight, and uλ ∈ V (λ) is the extremal weight vector of weight λ.