Symmetric semi-invariants for some Inonu-Wigner contractions-II-Case B even

Abstract

We consider a proper parabolic subalgebra p of a simple Lie algebra g and the Inonu-Wigner contraction of p with respect to its decomposition into its standard Levi factor and its nilpotent radical : this is the Lie algebra which is isomorphic to p as a vector space, but where the nilpotent radical becomes an abelian ideal of this contraction. The study of the algebra of symmetric semi-invariants under the adjoint action associated with such a contraction was initiated in my paper entitled : Symmetric Semi-Invariants for some Inonu-Wigner Contractions-I, published in Transformation Groups, January 2025, wherein a lower bound for the formal character of this algebra was built, when the latter is well defined. Here in this paper we build an upper bound for this formal character, when p is a maximal parabolic subalgebra in a classical simple Lie algebra g in type B, when the Levi subalgebra of p is associated with the set of all simple roots without a simple root of even index with Bourbaki notation (we call this case the even case). We show that both bounds coincide. This provides a Weierstrass section for the algebra of symmetric semi-invariants associated with such a contraction and the polynomiality of such an algebra follows.