Level Zero Fundamental Representations over Quantized Affine Algebras and Demazure Modules

Source abstract

Let W(\varpi_k) be the finite-dimensional irreducible module over a quantized affne algebra U'_q(\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^−(−Λ_0 + \varpi_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(\mathfrak g)(u_λ \otimes u_µ) → V(λ + µ) that sends u_λ \otimes u_µ to u_{λ+µ} . Here V(λ) is the extremal weight module with λ as an extremal weight, and u_λ \in V(λ) is the extremal weight vector of weight λ .