Symmetric Semi-invariants for some Inonu-Wigner contractions

Abstract: 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 $\mathfrak p$, we may consider 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. Then $\tilde\mathfrak p$ is a Lie algebra, which is a special case of In\"on\"u-Wigner contraction and may be considered as a degeneration of the parabolic subalgebra $\mathfrak p$. Let $S(\tilde\mathfrak p)$ be the symmetric algebra of $\tilde\mathfrak p$ (it is equal to the symmetric algebra $S(\mathfrak p)$ of $\mathfrak p$) and consider the algebra of semi-invariants $Sy(\tilde\mathfrak p)\subset S(\tilde\mathfrak p)$ under the adjoint action of $\tilde\mathfrak p$. Using what we call a generalized PBW filtration on a highest weight irreducible representation $V(\lambda)$ of $\mathfrak g$, induced by the standard degree filtration on the enveloping algebra $U(\mathfrak m^-)$ of $\mathfrak m^-$, the nilpotent radical of the opposite parabolic subalgebra $\mathfrak p^-$ of $\mathfrak p$, one obtains a lower bound for the formel character of the algebra $Sy(\tilde\mathfrak p)$, when the latter is well defined.