Introduction

Let t be an arbitrary symmetrizable Kac-Moody Lie algebra and Uq(t) the corresponding quantized enveloping algebra of t defined over C(q). One can associate to any dominant integral weight μ of t an irreducible integrable Uq(t)-module L(μ). These modules have many interesting properties and are well understood, [K], [L1].

More generally, given any integral weight λ, Kashiwara [K] defined an integrable Uq(t)-module Vmax(λ) generated by an extremal vector vλ. If w is any element of the Weyl group W of t, then one has V max(λ) ≅ V max(wλFurther, if λ is in the Tits cone, then V max(λ) ≅ L(w0λ), where w0 ≅ W issuch that w0</subλ is dominant integral. In the case when λ is not in the Tits cone, the module V max(λ) is not irreducible and very little is knowna bout it, although it is known that it admits a crystal basis, [K].

In the case when t is an affine Lie algebra, an integral weight λ is not in the Tits cone if and only if λ has level zero. Choose w0 ∈ W so that w0λ is dominant with respect to the underlying finite-dimensional simple Lie algebra of .t