Parabolic Adjoint Action, Weierstrass Sections and Components of the Nilfibre in Type $A$

This work is a continuation of [Y. Fittouhi and A. Joseph, Weierstrass Sections for Parabolic adjoint action in type $A$]. Let $G$ be an irreducible simple algebraic group and $B$ a Borel subgroup of $G$. Let $\mathfrak n$ be the Lie algebra of the nilradical of $B$. Consider an irreducible subgroup $P$ of $G$ containing $B$. Let $P'$ be the derived group of $P$. Let $\mathfrak m$ be the Lie algebra of the nilradical of $P$. A theorem of Richardson asserts that the algebra $\mathbb C[\mathfrak m]^{P'}$ of $P$ semi-invariants is multiplicity-free. A linear subvariety $e+V$ such that the restriction map induces an isomorphism of $\mathbb C[\mathfrak m]^{P'}$ onto $\mathbb C[e+V]$ is called a Weierstrass section for the action of $P'$ on $\mathfrak m$. Here in type $A$ such a section is constructed, but in better form than that given in Sect. 4, loc cit. Yet the main difference is a complete change of emphasis from the construction of a Weierstrass section, to its application. Let $\mathscr N$ be the nilfibre relative to this action. From the construction of a Weierstrass section $e+V$, it is shown that $e \in \mathscr N$. Then $P.e$ is contained in a unique irreducible component $\mathscr C$ of $\mathscr N$. The structure of $e+V$ is used to give a rather explicit description of $\mathscr C$ as a $B$ saturation set, that is of the form $\overline{B.\mathfrak u}$, where $\mathfrak u$ is a subalgebra of $\mathfrak n$ . This algebra is not necessarily complemented by a subalgebra in $\mathfrak n$ and so $\overline{B.\mathfrak u}$ is not necessarily an orbital variety closure (hence Lagrangian) but it can be. It is shown that $\mathscr C$ need not contain a dense $P$ orbit and this by a purely theoretical analysis. This occurs for an appropriate parabolic in $A_{10}$ and is possibly the simplest example. In this particular case $\mathscr C$ is not an orbital variety closure.