Kashiwara’s 1990 conjecture on B(λ) forming a Q-basis of L(λ)/vL(λ)

Establish that, for the quantized enveloping algebra U attached to a generalized Cartan matrix C and an integrable highest weight U-module L_λ, the set B(λ) consisting of the nonzero images of the explicitly defined collection X_λ under the quotient map L(λ) → L(λ)/vL(λ) forms a Q-basis of L(λ)/vL(λ), where L(λ) denotes the A-submodule of L_λ spanned by X_λ and A is the subring of Q(v) consisting of elements regular at v = 0.

Background

Appendix A.5 summarizes Kashiwara’s 1990 construction for an integrable highest weight module L_λ: an explicit set X_λ ⊂ L_λ, the A-submodule L(λ) spanned by X_λ, and the set B(λ) of nonzero images of X_λ in the quotient L(λ)/vL(λ), where A is the subring of Q(v) regular at v = 0. In [K90] it was conjectured that B(λ) is a Q-basis of this quotient.

The document further notes that this conjecture was proved for finite classical types in [K90] and subsequently announced in [K90a] and proved in full generality in [K91]. The quotation is included here because the paper explicitly states the conjecture while recounting the historical development.