On semi-invariants and index for biparabolic (seaweed) algebras, II

In a continuation of I, we study the semi-invariant subalgebra Sy(q) of S(q) for q a biparabolic subalgebra of a semisimple or of an affine Lie algebra. In most cases this is shown to be polynomial and we compute the degrees and weights of the generators. We describe explicitly the canonical truncation qE of q for which Sy(q)=SqE(qE).Let t denote the nilradical of q and q′ its derived algebra. We show that the subalgebra Sq′(t) of Sq′(q)=Sy(q) is always polynomial and describe the degrees and weight of the generators. Finally a sum rule is given for the “false” degrees of Sq′(q). These are true degrees in most cases.