The determination of character formulae is a basic goal in the theory of Kac-Moody Lie algebras and superalgebras. Notably the Weyl denominator formula which ensues from this work is one of the most important results in representation theory leading notably to extensions of classical results in number theory developed by Fermat, Gauss and Jacobi to study Waring's problem.
The first correct proof of the Demazure character formula valid for large parameters was found at the Weizmann Institute in 1995. It implies the normality of a Schubert variety (though not its deeper projective normality). This theory used what are called Joseph functors for which there are still important questions to be resolved. It led to our conjecture of the existence of Demazure flags. This was settled by Mathieu for the semisimple Lie algebra and later in (2005) at Weizmann for all simply-laced symmetrizable Kac-Moody algebras.
Character formulae have also an analogue in crystal theory. Borcherds had extended Kac-Moody theory to encompass a description of the Monster - the largest sporadic finite simple group. At Weizmann the Littelmann path model was extended to Borcherds-Kac-Moody Lie algebras and an entirely combinatorial proof of the deep Kashiwara involution was established. This had the advantage of also been valid in the non-symmetrizable case.
The theory of Lie superalgebras arose from the statistics of
particle physics. Kac gave a list of all Lie superalgebras
of "Kac-Moody" type. "Ghost centres" of these superalgebras were
described in Weizmann.
Vacuum modules are modules over special Kac-Moody algebras and superalgebras, which give rise to certain algebraic structures (W-algebras) appeared in conformal field theory coming from theoretical physics. In 2007-2009 Weizmann scientists in conjunction with Kac obtained a criterion of simplicity of vacuum modules; this result was an important step in studying the crucial question of the simplicity of a W-algebra. In 2010-2013 Weizmann scientists in conjunction with Kac proved character formulae for large class of vacuum modules over Lie superalgebras giving furthermore a host of new Weyl denominator formulae.