We provide several equivalent descriptions of a highest weight category using recollements of abelian categories. Also, we explain the connection between sequences of standard and exceptional objects.

We will now turn to the problem of classifying and constructing all finitedimensional representations of a complex semi-simple Lie algebra (or, equivalently, of a compact Lie group). It turns out that such representations can be characterized by their “highest-weight”. The first method we’ll consider is purely Lie-algebraic, it begins by constructing a universal representation with a given high...

Let G be a semisimple, simply connected algebraic group defined over an algebraically closed field k of positive characteristic p. Assume that p is larger than the Coxeter number of G and that the Lusztig character formula holds for regular restricted dominant weights. In this paper, we introduce two new highest weight categories C even and C reg odd , both full subcategories of the category C ...

with the left regular representation of G on L(G) corresponding to the action on the factor Vi and the right regular representation corresponding to the dual action on the factor V ∗ i . Under the left regular representation L (G) decomposes into irreducibles as a sum over all irreducibles, with each one occuring with multiplicity dim Vi (which is the dimension of V ∗ i ). To make things simple...