Let \(\mathfrak {p}\) be a proper parabolic subalgebra of a simple Lie algebra \(\mathfrak {g}\) . Writing \(\mathfrak {p}=\mathfrak r\oplus \mathfrak m\) with \(\mathfrak r\) being the Levi factor of \(\mathfrak {p}\) and \(\mathfrak {m}\) the nilpotent radical of \(\mathfrak {p}\) , the semi-direct product \(\tilde{\mathfrak {p}}=\mathfrak r\ltimes (\mathfrak {m})^a\) , where \((\mathfrak {m})^a\) is an abelian ideal of \(\tilde{\mathfrak {p}}\) , isomorphic to \(\mathfrak {m}\) as an \(\mathfrak r\) -module, is a Lie algebra. This is a special case of Inönü-Wigner contraction and may be considered as a degeneration of \(\mathfrak {p}\) . For any Lie algebra \(\mathfrak {a}\) , denote by \(Sy(\mathfrak {a})\) the algebra of symmetric semi-invariants in the symmetric algebra \(S(\mathfrak {a})\) of \(\mathfrak {a}\) under the adjoint action of \(\mathfrak {a}\) . In this paper we are interested in the polynomiality of the algebra \(Sy(\tilde{\mathfrak {p}})\) . Inspired by our method in Fauquant-Millet, F., Joseph, A. (Ann. Sci. Éc. Norm. Sup. 38, 155–191 2005) where we studied the polynomiality of \(Sy(\mathfrak {p})\) (the nondegenerate case), we obtain in this paper a lower bound for the formal character of the algebra \(Sy(\tilde{\mathfrak {p}})\) , when the latter is well defined. The method in the nondegenerate case does not apply directly in the degenerate case : in the present paper we define a so-called generalized PBW filtration on a highest weight irreducible representation of \(\mathfrak {g}\) to provide the lower bound. Combined with an upper bound we will construct in the near future for particular contractions \(\tilde{\mathfrak {p}}\) , our goal is to show that the algebra \(Sy(\tilde{\mathfrak {p}})\) is a polynomial algebra, by showing that both bounds coincide.
