#### Milestone Year

### 2000

#### Character Formulae

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.