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 B_J which as a set identifies with the free N module of rank |J| and showed that it contains a “highest weight” subcrystal B_J(∞) having some remarkable combinatorial properties. The goal of the present work is to exhibit B_J(∞) as an explicit polyhedral subset of B_J by constructing for each simple root α, a set of dual Kashiwara functions which are linear functions on B_J and whose maximum restricted to B_J(∞) 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.