Dual Kashiwara Functions for the B(∞) Crystal

Abstract

Let g be a Kac-Moody algebra. For each sequence J of reduced Weyl group elements, Kashiwara constructed a crystal BJ which as a set identifies with the free N module of rank |J| and showed that it contains a “highest weight” subcrystal BJ(∞) having some remarkable combinatorial properties. The goal of the present work is to exhibit BJ(∞) as an explicit polyhedral subset of BJ by constructing for each simple root α, a set of dual Kashiwara functions which are linear functions on BJ and whose maximum restricted to BJ(∞) determines the dual Kashiwara parameter \( \varepsilon _\alpha ^* \). Up to a natural conjecture concerning identities in the Demazure modules, it is shown that these functions are given through rather explicitly determined “trails” with respect to J in the fundamental module of lowest weight \( - \varpi _\alpha ^ \vee \) a for the Langlands dual of g. The proof uses Kashiwara duality extended to the non-symmetrizable case and the theory of S-graphs developed by the author.