Canonical Bases for the Miniscule Modules of the Quantized Enveloping Algebras of Types B and D

Let g be a finite-dimensional semisimple Lie algebra over C, and let U be the qanalogue of its universal enveloping algebra defined by Drinfel’d [3] and Jimbo [5]. According to [7, 3.5.6, 6.2.3 & 6.3.4], for each dominant weight λ in the weight lattice of g there is an irreducible, finite-dimensional highest weight U -module V (λ) with highest weight λ. Kashiwara [6] and Lusztig [7, 14.4.12] have independently shown the existence of a certain canonical basis B(λ) for V (λ). For 1 ≤ r ≤ rank(g) let ωr be the r-th fundamental weight (see §§2 and 3 for the numbering of the Dynkin diagrams we are using). Suppose that g is of type Bn−1 or Dn (n ≥ 4). In the former case, fix r = n− 1, and in the latter case, fix r ∈ {1, n − 1, n}. Let Vr be the fundamental U -module V (ωr). Then Vr is miniscule (see [1, VIII,§7.3]). Let W r be the set of distinguished left coset representatives in the Weyl group W of g with respect to the parabolic subgroup Wr generated by all of the fundamental generators s1, s2, . . . , sn of W except sr.