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Fix an irreducible (finite) root system
and a choice of positive roots. For any algebraically closed field
consider the almost simple, simply connected algebraic group
with root system
. One associates with any dominant weight
-modules with highest weight
, the Weyl module
and its simple quotient
be dominant weights with
$\mu <\lambda $
is maximal with this property. Garibaldi, Guralnick, and Nakano have asked under which condition there exists
is a composition factor of
, and they exhibit an example in type
where this is not the case. The purpose of this paper is to to show that their example is the only one. It contains two proofs for this fact: one that uses a classiffication of the possible pairs
$(\lambda ,\mu )$
, and another that relies only on the classiûcation of root systems.
In 1926 Hermann Weyl published a paper that contains his character formula for irreducible finite dimensional complex representations of complex and real semi-simple Lie groups and their Lie algebras. It can also be interpreted as a character formula for connected compact groups and for semi-simple algebraic groups in characteristic 0. (Here I am using modern terminology; when Weyl wrote his paper, terms like “Lie groups” were not yet in use.)
When we look at Weyl's character formula as a statement for Lie algebras, then it is a theorem on purely algebraic objects. However, Weyl used analytic methods to prove it. Not surprisingly, people looked for algebraic proofs. These attempts were finally successful and led also to useful reformulations of Weyl's formula. This development will be described in the first section of this survey.
The other topic to be discussed will be the search for analogues to Weyl's formula in more general cases. To start with, a finite dimensional complex semi-simple Lie algebra has an abundance of irreducible representations that are infinite dimensional. It was natural to look for character formulae for at least some families of representations sharing features of the finite dimensional ones — for example those generated by a highest weight vector.
Furthermore, it was also natural to go beyond finite dimensional complex semi-simple Lie algebras. There are several algebraic objects that share many structural features with these Lie algebras and that have similar representation theories.
Let [gscr ] be one of the Lie algebras [sscr ][Iscr ]n(K)
or [sscr ][oscr ]2n+1(K) over an algebraically
field K of characteristic p>0. Suppose in the first
that n∉Zp and in the second
case that p≠2. This assumption implies that [gscr ] is simple.
paper I study certain irreducible representations of [gscr ].