Polytopes for Crystallized Demazure Modules and Extremal Vectors

Demazure’s character formula for arbitrary Kac-Moody Lie algebra was given by S.Kumar and O.Mathieu independently ([6],[8]) by using geometric methods. In 1995, P.Littelmann gave some conjecture (partially solved by himself) about the relation between Demazure’s character formula and crystal bases [7], which was solved affirmatively by M.Kashiwara [3]. Then it gave purely algebraic proof for Demazure’s character formula for symmetrizable Kac-Moody Lie algebras. Here let us see those formulations. Let g be a symmetrizable Kac-Moody Lie algebra (in the context of “crystal base”, we need “symmetrizable”), and n be the nilpotent subalgebra of g. Furthermore, let Z[P ] be the group algebra of the weight lattice P and W be the Weyl group associated with g. Then Demazure operator Dw : Z[P ] −→ Z[P ] (w ∈ W ) is given as follows: for i ∈ I (index set) we set Di(e ) := e(1 − eii)/1− ei and for w = sil · · · si1 set Dw := Dil · · ·Di1 , which is well-defined. Let V (λ) be the irreducible highest weight module with the highest weight λ and uwλ be the extrmal vector with the weight wλ (w ∈ W ). Then, Demazure’s character formula is described as follows: ch(U(n)uwλ) = Dw(e ). (1.1)