Let W(ω_k_) be the ﬁnite-dimensional irreducible module over a quantized affne algebra U'q(g) with the fundamental weight ω_k_ as an extremal weight. We show that its crystal B(W(ω_k_)) is isomorphic to the Demazure crystal B_−(−Λ0 + ω_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 U'q(g)(_u_λ ⊗ _u_µ) → V(λ + µ) that sends _u_λ ⊗ _u_µ to _u_λ+µ. Here V(λ) is the extremal weight module with λ as an extremal weight, and _u_λ ∈ V(λ) is the extremal weight vector of weight λ.
Cite this article
Masaki Kashiwara, Level Zero Fundamental Representations over Quantized Affne Algebras and Demazure Modules. Publ. Res. Inst. Math. Sci. 41 (2005), no. 1, pp. 223–250DOI 10.2977/PRIMS/1145475409