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 Inönü-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.

38 pages