Level Zero Fundamental Representations over Quantized Affne Algebras and Demazure Modules

Abstract

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.