Dual Kashiwara Functions for the B(∞) Crystal

  • Anthony JosephEmail author
Part of the Progress in Mathematics book series (PM, volume 326)


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.


Crystals Kac-Moody algebras 

Mathematics Subject Classification (2010):



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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Donald Frey Professional Chair, Department of MathematicsThe Weizmann Institute of ScienceRehovotIsrael

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